I think it's important to interpret in light of context. For example, Marx in a letter to Engels complained about his close friend - as well as enemy - Ferdinand Lasalle, a Prussian jurist who he in quote crude language - and here I'm relying on a translation by Wikipedia on him: (hidden due to offensive language)

The J***** n***** Lasalle

The Jewish nigger Lasalle

Lasalle, though Jewish was not a negro. Obviously, he was expressing his animosity towards him in language that was then current. Racial sensitivities weren't as sensitive as they are now.

The question is whether this amounts to racism. Racism is not just about colour but also about the exercise of power through racial difference. There is nothing of that here. Marx had fallen out with Lasalle and was expressing his personal animosity - and not helping to construct an ideology of race - at least here.

The extracts come from the essay The Magyar Struggle, Friedrich Engels (8 January 1849) penned by Engels in the 194th issue of Neue Rheinische Zeitung a paper cofounded with Marx and which at its height had a circulation of around 3,000 subscribers. It is titled, The Magyar Uprising.

Magyars are ethnic Hungarians. This essay appeared in a sequence of articles during the Spring of Nations during the Revolutionary period of European history beginning in 1848. It all began in France, spread to Austria and then to Hungary as well as to other parts of Europe. A new liberal and democratic government was founded in Hungary led by the feted liberal Kossuth with the assent of the Austrian Emperor. However, a reaction set in with the Austrian Empire attempting to reign in his recalritant provinces, starting in Northern Italy, then Austria and then Hungary. It's at the cusp of this counter-revolutionary reaction that Engels is writing this polemical article.

Here, Engels is attempting to place this history and what to him would be contemporary events in an overarching philosophy of history. Marx's dialectical materialism is essentially such a history and is lightly touched on in this article.

However, when he says the Poles, the Germans and the Magyars are "revolutionary", he means only against the landed aristocracy and not as we would now understand it, as against Capital. In a sense, there is no difference. It's the first movement that set up liberal democracy in the West and a fulfillment of Enlightment ideals. Marx and Engels saw this movement continuing through socialism which would tame Capital ending finally in communism, the era after the death-rattle of Capital. None of that theory is touched upon here. They are focused on the first movement of an awakening of European conscienceness and which they are, breathlessly, in the middle of. It was a time of tulmult, a time of ferment. Whats happening in Europe today with Brexit and the war in Ukraine is as nothing compared to it. It was the founding of what we now know as The West.

When he writes:

All the other large and small nationalities and peoples are destined to perish before long in a world revolutionary storm.

He means what we would now term as being assimilated - not literally 'perish'. Of course today, we understand the virtue of diversity and difference is celebrated. Thos is an outcome of democracy but not an outcome Engels foresaw on the basis of what he writes here.

When he writes:

There is no country in Europe which does not harbour in some corner or other, one or several ruined fragments of peoples ... these relics of a nation mercilessly trampled underfoot in the course of history, as Hegel says, these residual fragments of peoples ...

He is talking not in terms of racism as understood today, an ideology of power and race, but in terms of Social Darwinism, this was actually despised by Darwin himself, but was extremely fashionable. It also formed a cornerstone of Nietzsche's philosophy including that of his untermensch and that of the Nazi's. It was fashionable in liberal opinion across much of the West, including the USA and Britain. It isn't exclusive to Engels. All this language disappeared after the Holocaust.

Again, an outcome of democracy - of what Engel is calling 'revolution' is a guaranteeing of minority rights. This was one of the early acts of the liberal Hungarian government.

Q: Were these two racist quotes written by Marx?

Short answer: no, neither is Marx the author, it was Engels writing from which some words were used for this, nor is that an authentic string of words even by Engels. And the original quote for this misquote is not racist. The misquote is pretty much racist, but neither is Marx the author, nor Engels. What Engels did write in the original and what was used to mangle this into the racist misquote is much more benign on that angle, although we do find at least some problematic sounding words in the article that is the original source for this misquote.

No. Not really. Based on fragments of real Marx/Engels quotes this is a fraudulent mash-up of a misquotation with mischievous invention mixed in. Perhaps the result of a long game of telephone that's not very old.

Both "quotes" are a far cry from what their source wrote, which is in both cases Engels, in an article about 'the Struggle of the Magyars' in the Neue Rheinische Zeitung newspaper of 1849.

The first "quote"

is a severely mangled derivation of this article written by Friedrich Engels:

Die ganze frühere Geschichte Östreichs beweist es bis auf diesen Tag, und das Jahr 1848 hat es bestätigt. Unter allen den Nationen und Natiönchen Östreichs sind nur drei, die die Träger des Fortschritts waren, die aktiv in die Geschichte eingegriffen haben, die noch jetzt lebensfähig sind - die Deutschen, die Polen, die Magyaren. Daher sind sie jetzt revolutionär.

Alle andern großen und kleinen Stämme und Völker haben zunächst die Mission, im revolutionären Weltsturm unterzugehen. Daher sind sie jetzt konterrevolutionär.

— Friedrich Engels: "Der magyarische Kampf", ["Neue Rheinische Zeitung" Nr. 194 vom 13. Januar 1849], in: "Karl Marx - Friedrich Engels - Werke", Band 6, pp165–173 (p168 for this quote), Dietz Verlag, Berlin/DDR 1959. online digitized from MEW originals

which is translated roughly:

All the past history of Austria proves it to this day, and the year 1848 has confirmed it. Among all the nations and little nations of Austria, there are only three who have been the bearers of progress, who have actively intervened in history, who are still viable now – the Germans, the Poles, the Magyars. Therefore, they are now revolutionary.

All other tribes and peoples, large and small, have first the mission to perish in the revolutionary world storm. That is why they are now counter-revolutionary.

or was translated 'officially' as:

All the earlier history of Austria up to the present day is proof of this and 1848 confirmed it. Among all the large and small nations of Austria, only three standard-bearers of progress took an active part in history, and still retain their vitality—the Germans, the Poles and the Magyars. Hence, they are now revolutionary.

All the other large and small nationalities and peoples are destined to perish before long in the revolutionary world storm. For that reason they are now counter-revolutionary.

— Karl Marx & Friedrich Engels: "Collected Works", International Publishers: New York, 1975 (Progress Publishers: Moscow, 1977). p230.

This is arguably sounding 'a bit strange', and even quite harsh in it terminology and phrasing, but much less eliminatory in intent than perhaps expected from a German author. He describes what he sees as more assimilation of former group identities into bigger currents of history, and he is not calling for speeding that up with anything, especially not killings or other physical measures. It is descriptive, not prescriptive.

The second "quote"

seems to be based on the same article, and thus was also written by Engels:

Es ist kein Land in Europa, das nicht in irgendeinem Winkel eine oder mehrere Völkerruinen besitzt, Überbleibsel einer früheren Bewohnerschaft, zurückgedrängt und unterjocht von der Nation, welche später Trägerin der geschichtlichen Entwicklung wurde. Diese Reste einer von dem Gang der Geschichte, wie Hegel sagt, unbarmherzig zertretenen Nation, diese Völkerabfälle werden jedesmal und bleiben bis zu ihrer gänzlichen Vertilgung oder Entnationalisierung die fanatischen Träger der Kontrerevolution, wie ihre ganze Existenz überhaupt schon ein Protest gegen eine große geschichtliche Revolution ist.

So in Schottland die Gälen, die Stützen der Stuarts von 1640 bis 1745.

So in Frankreich die Bretonen, die Stützen der Bourbonen von 1792 bis 1800.

So in Spanien die Basken, die Stützen des Don Carlos.

So in Östreich die panslawistischen Südslawen, die weiter nichts sind als der Völkerabfall einer höchst verworrenen tausendjährigen Entwicklung. Daß dieser ebenfalls höchst verworrene Völkerabfall sein Heil nur in der Umkehr der ganzen europäischen Bewegung sieht, die für ihn nicht von Westen nach Osten, sondern von Osten nach Westen gehen sollte, daß die befreiende Waffe, das Band der Einheit für ihn die russische Knute ist - das ist das Natürlichste von der Welt.

— pp172–173.

translated quickly as:

There is no country in Europe that does not possess in some corner one or more ruins of peoples, remnants of an earlier population, pushed back and subjugated by the nation that later became the bearer of historical development. These remnants of a nation mercilessly trampled underfoot by the course of history, as Hegel says, these wastes of peoples* [see explanation below, LLC], become each time and remain until their complete extermination or denationalisation the fanatical bearers of the counter-revolution, just as their whole existence is in itself a protest against a great historical revolution.

Thus in Scotland the Gaels, the supporters of the Stuarts from 1640 to 1745.

So in France the Bretons, the supporters of the Bourbons from 1792 to 1800.

So in Spain the Basques, the supporters of Don Carlos.

So in Austria the Pan-Slavic Southern Slavs, who are nothing more than the residual of a highly confused thousand-year development. That this likewise highly confused residual sees its salvation only in the reversal of the whole European movement, which for it should go not from West to East but from East to West, that the liberating weapon, the bond of unity for it is the Russian knout - that is the most natural thing in the world.

translated 'officially'

There is no country in Europe which does not have in some corner or other one or several ruined fragments of peoples, the remnant of a former population that was suppressed and held in bondage by the nation which later became the main vehicle of historical development. These relics of a nation mercilessly trampled under foot in the course of history, as Hegel says, these residual fragments of peoples always become fanatical standard-bearers of counter-revolution and remain so until their complete extirpation or loss of their national character, just as their whole existence in general is itself a protest against a great historical revolution.

Such, in Scotland, are the Gaels, the supporters of the Stuarts from 1640 to 1745.

Such, in France, are the Bretons, the supporters of the Bourbons from 1792 to 1800.

Such, in Spain, are the Basques, the supporters of Don Carlos.

Such, in Austria, are the pan-Slavist Southern Slavs, who are nothing but the residual fragment of peoples, resulting from an extremely confused thousand years of development. That this residual fragment, which is likewise extremely confused, sees its salvation only in a reversal of the whole European movement, which in its view ought to go not from west to east, but from east to west, and that for it the instrument of liberation and the bond of unity is the Russian knout — that is the most natural thing in the world.

— Karl Marx & Friedrich Engels: "Collected Works", International Publishers: New York 1975 (Progress Publishers: Moscow, 1977). p234–235.

A complete English translation online:

— "The Magyar Struggle", Author(s) Friedrich Engels, Written 8 January 1849, Source: Marx-Engels Collected Works, Volume 8, p. 227; First published: in Neue Rheinische Zeitung No. 194, January 13, 1849. Collection(s): Neue Rheinische Zeitung. Keywords : Hungary, National question, Pan-Slavism, Revolutions of 1848, Terror

The source for the actual Engels quotes

Since the quote's source Engels in MEW was disputed in comments for authenticity and authoritativeness:

The authoritative reference given is the printed book, with all necessary details.

The online version linked is not the authoritative source, but just linked here for your convenience. And it is matching the printed book. Want proof for that as well?

Here come the scans of that volume for the first page and those containing the quotes:

(click for large version)

Alternatively, @Mark Johnson linked this PDF for a complete volume…

The context for the original newspaper article would be found in archives like this one here:

— Deutsches Textarchiv: Neue Rheinische Zeitung — Organ der Demokratie. No 194. Köln, Samstag den 13. Januar. 1849.

Only that this very article is not present at that site, scheduled for re-publication in MEGA2:

Edition: [Friedrich Engels: Der magyarische Kampf, vorgesehen für: MEGA2, I/8.]

It will be available at MEGAdigital, the (2nd) successor to MEW, edited by Internationale Marx-Engels-Stiftung and also published with German federal funds at the Berlin-Brandenburgischen Akademie der Wissenschaften (Berlin-Brandenburg Academy of Sciences and Humanities). Mega I/8 is since 2020 in printed book form or PDF available at doi:10.1515/9783110683394, the article about 'the Magyar's struggle' with historical critical commentary and apparatus at doi:10.1515/9783110683394-016.

Meaning of the actual quotes

For interpreting — or for starters adequately translating — the most controversial word actually used by Engels, Völkerabfälle, — translated 'roughly' above as 'wastes of people' and 'residual' (in the 'official translation always "residual fragments of people") from German Volk people & Abfall (link1, link2), this is helpful:

Yet, The Soviet Story’s criticism of Engels and the moral bankruptcy of the latter’s scathing remarks on the so-called Völkerabfälle has been challenged by historians. The English translation provided for the word by Watson in the film is indeed deceiving – “racial trash” does not quite capture the original meaning of the word as used by Engels in his quoted article of 1849 in Neue Rheinische Zeitung. A more precise translation would be “residual nations” or “refuse of nations,” that is, those left behind, or discarded, by the dominant civilizations. The synonyms in the political lexicon of Engels, used in parallel to the Völkerabfälle are more illuminating here: Völkertrümmer (the left-overs of nations), or Völkerruinen (the ruins of nations) imply the left-overs of previously existing, then disappeared nations, or the remnants of the great historical nations (e.g., southern Slavs which had formerly lived in Austrian territory). The issue for Engels was really about whether these nations were sufficiently strong for establishing their own nation-states, not their destruction. Engels rather pointed to the inevitable assimilation of the stateless nations in case they were unable to construct the states of their own, but he did not argue for their physical destruction as claimed by Watson in the film.

— Maria Malksoo: "A Baltic Struggle for a "European Memory": The Militant Mnemopolitics of The Soviet Story", Journal of Genocide Research, Vol 20, No 4: Special issue on The Holocaust/Genocide Template in Eastern Europe, pp 530–544, 2018. PDF/doi

The provenance of the mangled quotes presented in question

The controversial word 'Völkerabfälle' appears exactly one time searching the MEW via mlwerke. The authorship, 'publication details' and date of this often copied internet dreck are off. They are off even in some collections —that quote-mine Marx/Engels for 'bad words' and/or passages to prove something— that otherwise are happy to copy more precise (looking) references that for example point to the exact page numbers in MEW.

The very wording of the second 'quote' "Man solle…" should give German speaking readers a hint. The verb used indicates indirect speech, meaning here that 'Marx' paraphrases/quotes some else in that sentence. His 'own' hypothetical appellative would be 'sollte'.

Google Books does not index this phrase as misquoted on Pluspedia, nor does archive.org. The regular web search engines seem to now only start to see this misquote version apparently after 2013. And very seldom do any of the hits produced for that even start to look reliable: comments, forums, blogs, facebook, twitter…