Consider that Charles Frederick, Duke of Holstein-Gottorp and the one that all Russian emperors after Catherine were descended from, would have been king of Sweden had his aunt not stolen the throne from him. Was there ever a time when the Tsars could have either attempted to take the Swedish throne or at the very least put a Holstein-Gottorp-Romanov on it?
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.
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.
Did the Russian Empire have a claim to Sweden? Was there ever a time where they could have pursued it?
Consider that Charles Frederick, Duke of Holstein-Gottorp and the one that all Russian emperors after Catherine were descended from, would have been king of Sweden had his aunt not stolen the throne from him.
The Holstein-Gottorp men had bad luck with women. Charles Fredrick son Peter the III of Russia was himself deposed and killed by his wife Catherine II, or Catherine the Great after only in power for 6 months.
Was there ever a time when the Tsars could have either attempted to take the Swedish throne or at the very least put a Holstein-Gottorp-Romanov on it?
Arguable not a Holstein as they supposedly only reigned in Russia for 6 months. Peter III was the first and last Holstein. Paul I who succeeded Catherine the Great, Catherine maintained was her son by extra marital affair with Sergei Saltykov and not her Husband Peter III who she dethroned to take power.
Yes, The Czars attempted it a few times. In the Russo-Swedish Wars of 1495-97 Russia formed an alliance with Denmark to replace the King of Sweden in exchange Russia was to received parts of Norway. They succeeded but the new King of Sweden craw fished and refused Russia it's bootie. In the Great Northern War(1700-21) Peter the Great successfully displaced the Swedish Empire as the supreme power in northern central Europe and took the land which is now Saint Petersburg as part of his bootie. In the Finnish War of (1808-1890) Russia split off the eastern third of now weakened Sweden and established an autonomous Grand Duchy of Finland, which Russia controled.
For most of Russia's history they were a country devoid of a reliable shipping port in which to carry on trade. In the east their primary port was Arkhangelsk on the white sea, frozen in much of the year. They had no north sea port. In the east their ports on the black sea were contained by the Ottoman empire which control over the Dardanelles Strait from the early 1400's made the black sea their lake. Russia long desired reliable year round ports. That was the reason for many of their wars with the Swedish Empire which controlled the north Sea as well as the Baltic States which controlled access to the North Sea. Peter the Great was the Russian Czar who obtained ports on both sea's for Russia in the late 16 and early 1700's.
Peter the Great
To improve his nation's position on the seas, Peter sought to gain more maritime outlets. His only outlet at the time was the White Sea at Arkhangelsk. The Baltic Sea was at the time controlled by Sweden in the north, while the Black Sea and the Caspian Sea were controlled by the Ottoman Empire and Safavid Empire respectively in the south.
- Swedish–Novgorodian Wars A series of conflicts between the 12th and 14th centuries.
- Russo-Swedish War (1495–97) Result of an alliance between Ivan III of Russia and Hans of Denmark.
- Russo-Swedish War (1554–57) Prelude to the Livonian War.
- Livonian War (1558–82) Fought for control of Old Livonia in the territory of present-day Estonia and Latvia.
- Russo-Swedish War (1590–95) Instigated by Boris Godunov in the hope of gaining the territory of the Duchy of Estonia.
- De la Gardie Campaign (1609-1610) A military campaign to put Vasili IV on the Russian throne. Part of the Polish–Muscovite War (1605–18). The campaign can be considered a prelude to the Ingrian War.
- Ingrian War (1610-17) Including an attempt to put a Swedish duke on the Russian throne.
- Russo-Swedish War (1656–58) Part of the Second Northern War.
- Great Northern War (1700–21) Conflict in which a coalition led by the Tsardom of Russia successfully contested the supremacy of the Swedish Empire in northern Central Europe and Eastern Europe.
- Russo-Swedish War (1741–43) Also known as the Hats' Russian War.
- Russo-Swedish War (1788–90) Also known as Gustav III's Russian War in Sweden, and Catherine II's Swedish War in Russia.
- Finnish War (1808–1809) Resulted in the eastern third of Sweden being established as the autonomous Grand Duchy of Finland within the Russian control