Did the Russian Empire have a claim to Sweden? Was there ever a time where they could have pursued it?

Technically the question should be: "Did any Russian Tsars or Emperors have a claim to be king of Sweden?" since claims to the throne of a kingdom are genealogical.

As it happens, I created several threads in the Historum site tracing the heirs of various kingdoms.

One of those threads lists the heirs of various individual kings and dynasties of Sweden down to the present:

https://historum.com/threads/heirs-of-the-kingdom-of-sweden.171986/1

If my research is correct, I will have listed the correct heirs. So checking to see if any kings of dynasties of Sweden have Russian Tsars or Emperors among their heirs is a way for you to find out.

In Post number 11 on page 2 I trace the heirs of King Charles XI Vasa of Sweden down to the Russian imperial house of Romanov-Holstein-Gottorp beginning with Emperor Peter III.

https://historum.com/threads/heirs-of-the-kingdom-of-sweden.171986/page-2

Of course there is the speculation that Empress Catherine Ii's son Emperor Paul was not the son of Peter III but of Catherine's lover. But apart from that speculation, every Emperor of Russia from 1762 and from 1796-1917 was the rightful genealogical heir of King Charles XI Vasa of Sweden and all the monarchs of Sweden from 1604-1720,

Post number 28 on page 3 traces a line of possible heirs of King Olof Skotkunung of Sweden (reigned c.995-c.1022) down through various princes of the Rurik dynasty to the mother of Tsar Michael Romanov, the first Romanov ruler and then down to Peter III and his descendants as in post number 11.

https://historum.com/threads/heirs-of-the-kingdom-of-sweden.171986/page-32

Post number 33 on page 4 traces a line of heirs of the Folkung or Bjalbo dynasty of Sweden that reigned from 1250 to 1364 down to the Vasa dynasty and thus down to Peter III of Russia and his Romanov heirs.

https://historum.com/threads/heirs-of-the-kingdom-of-sweden.171986/page-43

Post number 51 & 52 on page 6 list the heirs of Svantepolk Knutsson Lord of Viby and his Wife Benedikte Sunnadotter of Ymseborg thrugh their daughter Ingeborg. Post number 53 traces their heirs through their daughter Ingrid down. Post number 54 traces their heirs through their daughter Ingrid by a different line of descent down to the Vasa dynasty of Sweden and down to Emperor Peter III of Russia.

https://historum.com/threads/heirs-of-the-kingdom-of-sweden.171986/page-64

And why shoudl anyone care who is the genealogoical heir of Heirs of Svantepolk Knutsson Lord of Viby and his Wife Benedikte Sunnadotter of Ymseborg?

Svantepolk Knutsson (died 1310), Lord of Viby in Östergötland, was a Scandinavian magnate.1

Svantepolk's father was Canute, Duke of Revalia, a bastard son of king Valdemar II of Denmark with a high-born Swedish lady Helena, daughter of Earl Guttorm. His mother had been a Pomeranian lady whose first name and precise origin is not known to us, but who is indicated to have come from the ducal family of Pomerelia. Svantepolk's brother was Eric, Duke of Halland who died in 1304.

Svantepolk settled in Sweden in the mid-13th century. His wife was Benedikte of Ymseborg, daughter of (Earl) Sune Folkesson, a grandson of earl Birger Brosa. Benedikte's mother was Helena Sverkersdotter, the daughter of king Sverker II of Sweden and his first wife Benedikte Ebbesdatter of the Galen family of Denmark. Benedikte's elder sister Catherine of Ymseborg was wife of king Eric XI of Sweden (1216–50), so Svantepolk was the late king's brother-in-law.

https://en.wikipedia.org/wiki/Svantepolk_of_Viby5

So Benedikte Sunnadotter of Ymseborg was possibly the heiress of the Sverker dynasty.

And in fact the previously mentioned post number 33 on page 4, which is about tracing the heirs of the Folkung or Bjalbo dynasty, includes as person number 2 in the line of descent "Bengt or Benedikte Sunesdotter married Svantepolk Knutsson" And it does trace their line o fheirs down to the Vasa Kings o fsweden and to the Romanovs beginning with Peter III. Though posts numbers 34, 35, & 36 trace alternate lines of heirs from Benedike Sunnadottoer of Ymseborg.

So Thus the Romanov-Hostein-Gottorp dynasty descended from Peter III could claim to be the rightful heirs of the Kings of Sweden of the Vasa and Palatinate-Zweibrucken dynasties from 1604-1720, and the possible heirs of the House of Munso, the first historical dynasty of Sweden, that ruled from about 970 to 1060, and possible heirs of the Folkung or Bjalbo dynasty of Sweden that reigned from 1250 to 1364, and possibly heirs of the House of Sverker which ruled Sweden on and off from 1130 to 1222.

So the dynasty of Romanov-Holstein-Gottorp certainly could make claims to be the rightful genealogical heirs of some Swedish kings and dynasties. Of course the validity of every such claim depended on Emperor Paul I being the son and thus heir of Emperor Peter III.

Yes, they did put a Holstein-Gottorp on the throne. But that didn't really help relations very much.

Following the disaster of the Great Northern War, Russia was the dominant power in Northern Europe. In the Treaty of Nystad, Sweden had to give up all the eastern Baltic dominions. More to the point of the question, Russia was also made a warrantor of the new Swedish constitution, which put most of the legal and executive power in the hands of the Swedish diet, the Riksdag. This smarted, but for a long while, the dominant Swedish politicians realised that going to war with Russia was not a good idea.

In 1741, a new generation of politicians had taken control, and wanted revenge. It ended terribly: in the war of 1741-1743 (a side-show to the war of Austrian succession; the Swedish politicians were not totally suicidal), Russia occupied all of Finland. Sweden lost a bit of territory, but more to the point of the question, the new Russian empress Elizabeth put her relative Adolf Frederick on the Swedish throne.

Adolf Frederick was a rather peaceable, quiet sort, who did not go much into politics, but his son Gustav III was more active, and would eventually fight a war with Russia. This ended in a draw, but at least cancelled Russias formal right to interfere in Swedish politics. Gustav's son in turn was Gustav IV Adolf, a staunch, even stubborn, opponent of Napoleon, which turned out to be a disastrous policiy when Russia switched sides. He ended up losing first Finland, and then his throne. He was first briefly replaced by his uncle Charles XIII, and then later by Charles XIV John, formerly one of Napoleon's marshals, who was adopted by Charles. At this point, I guess Russia could have protested, but the politics had shifted once again, and since both Sweden and Russia was now opposed to Napoleon, rocking the boat would have been a bad idea.

For the rest of the 19th century, Russia and Sweden had little reason for conflict. Charles XIV John was again a realistic sort of person when it came to foreign politics, as well as the politicians that followed him. Russia also had a pretty good set up, with Finland as a not too troublesome province and buffer.

So, to conclude: in the 18th century, Russia did put someone on the Swedish throne, and had some legal rights to interfere in Swedish politics, but never anything beyond that.

Absolutely not. Through the entire 17th century and most of the 18th Sweden was the dominant military power across the Baltic Sea, a significantly stronger military power than Russia. Not until the end of the 18th Century is Russia approaching Sweden in military strength. It is not until 1809 that Russia is even strong enough to wrest the bulk of Finland from Sweden in the Finnish War.

After this point Russia has secured its dual northern interest in the port of St. Petersburg and a minimization of its border across the base of the Scandinavian Peninsula.

From a strategic standpoint this allows it to concentrate on its strategic interests in the Black Sea coast, Crimea, and the Balkan Slavic countries.