Did Stalin want to restore the Russian Empire's borders?

Lev, you apparently hold the assumption that post-WWII borders were "defined" by Molotov-Ribbentrot pact. USSR had to claim at least that territory, but when Germany became a war loser any winning coutry could take a piece of a losing country or their ally. How exactly it happened, can be read in Churchill WWII autobiography, and in protocols of Yalta and Potsdam conferences.

Lvov transfer,I beleive, was result of one-to-one talk between Churshill and Stalin account of which I read in Churchill autobuigrahy. Churshill and Stalin, on one-to-one, argeed to move poland to the west, giving piece of German territory to Poland (with subsequent transfer of German population), plus giving piece of Poland territory to Russia. The talk took only two minutes, both parties having no disagreements, and not a word was exchanged

Churchill describe this talk in details.

Churchill writes (I write from memory): "I took a piece of paper and made a sketch of boundaries of Poland and Gernany. Stalin watched me. Then I draw a line on German territory to the west of Poland. Then I draw another line to the west of Russian-Polish territory. I showed this to Stalin. Stalin nodded." That was it. (Churchill draw line along large rivers that made his sket precise, actually. I forget river names).

Boundaries and spheres of influences followed from the positions taken by the troops on the ground, and then agreements between CHurchill and Stalin. This is obvious from protocols of Yalta and Potsdam conferences.

Stalin want "secure" borders for Russia. To the south, that meant parts of Turkey, at least the "Straits," and part of the Balkans, say Bulgaria to secure them, plus parts of Persia, all the way to the Persian Gulf, according to sources such as the Molotov-Schulenburg talks, quoted in sources like William L. Shirer's "The Rise and Fall of the Third Reich." These were far beyond Russian imperialist borders.

To offset this, Stalin was willing to accept "less" in the west than the Russian borders. He was willing to "horse trade" parts of Poland, e.g. Warsaw, and even LIthuania held by the Russian Czars. What he wanted most was the Pripet Marshes, and the forest along and in front of them, for defense.

The Soviet Union did invade Poland in February 1919 – March 1921 with the war aims of reconquest and spread of Communism -- which as comments have pointed out was the main reason (read: excuse) for Soviet expansion. The Soviet union's wars and assimilations of the kahnates during Lenin set a policy of conquest via politics and military might. Stalin was following and using Lenin to justify his own cult. So, I would yes that Stalin wanted indeed to revive the Russian Empire and expend it to cover the Earth.

Source: Setting the East Ablaze: Lenin's Dream of an Empire in Asia by Peter Hopkirk, God's Playground A History of Poland: Volume II: 1795 to the Present: 1795 to the Present Vol 2 by Norman Davies and White Eagle, Red Star: The Polish-Soviet War 1919-20 by Norman Davies

There is nothing special about borders of former Russian Empire when considering Soviet expansion.

There is an official USSR diplomatic note to Germany (Molotov to Schulenberg on 25 November 1940). It is a conditional agreement of USSR to join Axis. The conditions were to extend USSR sphere of influence to: Bulgaria, Bosfor, Dardanelles, and south of Batumi and Baku towards Gulf of Persia (source: Mark Solonin "25 June" p. 227, citing the Russian Presidential Archive set 3/64, doc 675 p. 108). These territories were obviously not in the former Russian Empire. Molotov sent a note acting on Stalin's orders.

"Did Stalin want..." What person X wanted or what person Y was thinking is not a good subject of historical research; our motives and desires are mysterious even to our close ones, so how could historians seventy years later know better?

What we know is that USSR did not hide from Germany their interest in expanding to these territories, and I think this is what your question is about, right?

Also, in 1945, Stalin negotiated with Allies to officially take a part of German Prussia (Koenigsberg, now Kaliningrad) - it was never a part of a former Russian Empire. (Unofficially, Stalin had much control over East Germany, Czechoslovakia, Hungary, Bulgaria, etc. These were never parts of a former Russian Empire.)

First of all, I don't really see why you think that USSR didn't get Lvov according to the pact. It is true that Germans captured Lvov before going back and letting Russians in (according to a statement by somebody who was in Lvov at that time, I didn't bother looking up online sources on that) but that appeared to be simply a misunderstanding. The German and Russian troops even held a joint parade in Lvov, no sign of disagreements.

Stalin's actions in 1939 and 1940 were indeed targeted at restoring the territory of USSR to the extents of the Russian Empire: invasion of Poland, then Winter War against Finland in which USSR captured Karelia (probably less than originally intended, all of Finland used to be Russian province), finally recapturing of Bessarabia and the Baltic states. All of these territories used to belong to the Russian Empire and were given away in the Treaty of Brest-Litovsk. So getting them back was well-received in the population.

But I wouldn't go as far as to claiming that Stalin wanted to revive the Russian Empire - monarchistic ideas in the USSR were a hard sell. The Soviet regime was always expansionist, with the stated goal of a world-wide revolution. It was a good opportunity to get territories back that were "rightfully theirs", so they did it. After World War II USSR expanded its control to a large part of Europe - these countries never belonged to the Russian Empire but given that they already were occupied it would have been stupid to simply give them up.