Traveling with Erlaubnisfiktion

As a United States citizen, that does not require a short-term visa for the Schengen Area, you are allowed 90 days in any 180-day period.

Once the Fiktionsbescheinigung as Fiktiv erlaubter Aufenthalt [Erlaubnisfiktion] (§ 81 Abs. 3 Satz 1 AufenthG) [initial application of a residence permit] has been issued, starting on that date, the days in Germany do not count towards the 90 days.

Since you don't need a visa, you may visit other Schengen Countries as long as the 90 days in any 180-day period has not been exceeded (the days outside of Germany, after the Fiktionsbescheinigung was issued, will count towards this sum of 90).

The amount of days is not set to 0 because of the issued Fiktionsbescheinigung.

Notes:
For persons that require a visa, a Fiktionsbescheinigung as Fiktiv erlaubter Aufenthalt is not considered a resident permit that exempts you from the visa requirement.

The quote of General Administrative Provisions for the Residence Act of section 81.3.6 (page 335) in Tor-Einar Jarnbjo answer is not compleate.

It continues with:

Vorschriften, wonach die Einreise ohne Aufenthaltstitel aus anderen Gründen zulässig ist (etwa Artikel 20 oder 21 SDÜ, Artikel 1 Absatz 2 der Verordnung (EG) Nummer 539/ 2001; §§ 16 bis 30 sowie 41 AufenthV), bleiben unberührt. Um den Unterschied zu verdeutlichen, sollte auf dem Trägervordruck von Fiktionsbescheinigungen, die nach Absatz 3 ausgestellt werden, der Vermerk „Gilt nicht für Auslandsreisen“ aufgenommen werden.

Regulations according to which entry without a residence permit is permitted for other reasons (e.g. Article 20 or 21 SDÜ, Article 1 Paragraph 2 of Regulation (EC) Number 539/2001; Sections 16 to 30 and 41 AufenthV) remain untouched. In order to make the difference clear, the note “does not apply to trips abroad” should be included on the carrier form of fiction certificates issued in accordance with paragraph 3.

For persons who do not require a Schengen Visa to enter the Schengen Area:.

Article 1 Paragraph 2 of Regulation (EC) Number 539/2001

Nationals of third countries on the list in Annex II shall be exempt from the requirement set out in paragraph 1 for stays of no more than 90 days in any 180-day period.

The Fiktionsbescheinigung as Fiktiv erlaubter Aufenthalt [Erlaubnisfiktion] (§ 81 Abs. 3 Satz 1 AufenthG) was issued base on:

§ 41 AufenthV
(1) Staatsangehörige von Australien, Israel, Japan, Kanada, der Republik Korea, von Neuseeland, des Vereinigten Königreichs Großbritannien und Nordirland im Sinne des § 1 Absatz 2 Nummer 6 des Freizügigkeitsgesetzes/EU und der Vereinigten Staaten von Amerika können auch für einen Aufenthalt, der kein Kurzaufenthalt ist, visumfrei in das Bundesgebiet einreisen und sich darin aufhalten. Ein erforderlicher Aufenthaltstitel kann im Bundesgebiet eingeholt werden.

(1) Citizens of Australia, Israel, Japan, Canada, the Republic of Korea, New Zealand, the United Kingdom of Great Britain and Northern Ireland within the meaning of Section 1 (2) number 6 of the Free Movement Act / EU and the United States of America can also apply for a stay who is not a short stay can enter and stay in the federal territory without a visa. A required residence permit can be obtained in Germany.

and therefore should not contain the note “Gilt nicht für Auslandsreisen”.

Can I travel within the Schengen zone whenever I like? I was under the impression that there were no border checks within Schengen

There are no border checks, but the 90 days in any 180-day period rule still applies, but is not strictly enforced for residence permit holders.

With a valid Fiktionsbescheinigung as Fiktiv erlaubter Aufenthalt (and later residence permit) you can always return directly to Germany. Entry through another Schengen country will also pose no problem, since they won't know you went to Hungary after the Fiktionsbescheinigung was issued.

Germany issues three different kinds of fictional certificates and the answer to your questions depends on which type you have. Based on your circumstances (waiting for the decision upon your first appplication for a residence permit), you will have been issued a fictional certificate based on § 81 Abs. 3 Satz 1 AufenthG. You will find a corresponding notice on page 3 of your certificate.

1. No. A fictional certificate based on § 81 Abs. 3 Satz 1 AufenthG does only grant you the right to stay in Germany until your application for a residence permit has been decided upon. You are right that there are usually no border checks when travelling within the Schengen area, but that does not necessarily mean that everyone is entitled to cross at an unstaffed, seemingly open border crossing. Both because of the covid situation and also still because of the 'refugee crisis' in 2015, immigration checks are much more common than they used to be on intra-schengen borders. Since you are planning to go to Hungary, all land border crossings from Hungary towards Austria are for example now permanently staffed. When crossing that border by car, I have some times been waved through, but also been stopped for a document check.

2. At least not because of the fictional certificate. It does not give you a right to reenter Germany if you decide to leave. If you are allowed to reenter, will in your situation depend on a fine count of the days you have already stayed in the Schengen area. There is no 'reset' of the 180 days as you seem to believe, but you are only allowed to stay 90 days within any floating period of 180 days. If you stayed 14 days in September, reentered on October 1st and for example leaves Germany on December 15th for your Christmas holidays, that will make up 90 days. You will then not be allowed to reenter Germany (or any other Schengen country) until 180 days after your entry in September. Depending on when you exactly entered in September, that will not be until late February or some time in March.

Addition: I am not going to participate in another fruitless discussion with Mark Johnson, I just wanted to explain why his answer is wrong. You can in the 'General Administrative Provisions for the Residence Act' (only available in German), in section 81.3.6. find the relevant clarifications regarding your fictional certificate. Here, it is clearly explained, that your fictional certificate does not count as a replacement for any of the credentials you are, according to the Residence Act section 4, required to be in posession of when entering Germany, among them are national visa, temporary and permanent residence permits. And to quote the relevant conclusion:

Anders als die Fiktionsbescheinigung, die nach Absatz 4 erteilt wird, ermöglicht die Fiktionsbescheinigung nach Absatz 3 daher keine Einreise in das Bundesgebiet.

Or my translation:

Unlike fictional certificates, which are issued based on section 4, the fictional certificates issued based on section 3 (my remark: § 81 Abs. 3 Satz 1) do therefore not allow you to enter the federal territory.

For the counting of 90 days to be interrupted while you are within the Schengen area, you need to be in posession of a national visa or a residence permit and be present in the country, which issued these documents. Since, explained in the provisions I referred to, Germany does not consider your fictional certificate as a replacement for a national visa or a residence permit, staying in Germany with a fictional certificate based on § 81 Abs. 3 Satz 1 does not interrupt the 90 days clock.

The differences between fictional certificates issued based on section 3 and section 4 are further explained in the provisions in paragraph 81.5.3:

Die nach Absatz 4 ausgestellten Fiktionsbescheinigungen berechtigen anders als die nach Absatz 3 ausgestellten Bescheinigungen in Verbindung mit einem anerkannten und gültigen Pass oder Passersatz zur Einreise in das Bundesgebiet und nach Artikel 21 SDÜ zu Reisen innerhalb des Schengen-Raums.

My translation:

Fictional certificates issued based on section 4, unlike fictional certificates issued based on section 3, do allow in connection with a valid and recognised passport or passport replacement to enter the federal territory and, according to article 21 of the Convention implementing the Schengen Agreement, to travel within the Schengen area.

And as xngtng also pointed out, not only the provisions to German law explain this, but also the Schengen Borders Code mandate, that fictional certificates issued based on section 3 do not interrupt the 90 days counter. The relevant part can be found in Article 6 (2) of the Schengen Borders Code: 'Periods of stay authorised under a residence permit or a long-stay visa shall not be taken into account in the calculation of the duration of stay on the territory of the Member States.'. And in Article 2 (16) we find that the definition of a 'residence permit' explicitely excludes 'temporary permits issued pending examination of a first application for a residence permit', which is exactly what your fictional certificate is.