I am seriously underinformed here - at what point of time did the Swedish navy lose its status as a force to reckon with?
Would you check history of last years of the Northern War, you would discover that Russians largely eliminated the Swedish naval threat (at least as far as Finnish coast-Aland route to Swedish shores is concerned) by 1719. Russian galley fleets landed thousands of infantry and Cossack (with their horses, mind you) troops plus artillery (effectively, small armies) on Swedish coast. It happened barely 15 years after Russians, who didn't have any naval cadre to speak of in the beginning, got access to Baltic coast. IMHO it proves that 10-15 years of very energetic effort, buttressed by existing shipbuilding capacity of Russian Baltic coast, could secure safe road for Russian invasion force to Swedish coast at any point after 1720.
I assume that taking Sweden would alienate Prussia away from Russia (and into the British camp, where there are also historical ties of an alliance) in the time afterwards
Agree. Prussians would not be thrilled. However, if they're afraid of Russians, there's long land border to worry about and Russian cossacks are way scarier than Russian sailors anyway. Besides, don't forget that all Prussian experiences to fight Russia pre-1914 were disasters (Russians occupied Berlin during 7 years war and Prussian corps dragooned by Napoleon suffered miserable fate).
But the Germans would, if needed, reinforce the Swedish as they reinforced the k.k.-Forces. It would be worth the reserves as the Russian army needs to distract forces from the main front as well.
Austrian experience was exactly the reason for my doubts. Despite all German reinforcements, Russians still owned more Austrian territory in January 1917 than Austrians owned Russian (I'm too lazy to check but I'm not sure Austrians had any at all). So, it is far from certain that Swedes would do any better. There're couple of other points to consider:
1. Germany would need to pull troops from frontline to send reinforcements to Sweden, it didn't have any spare troops. On the flip side, Russia did keep tens of thousands troops in Finland during the war, so it would be less distraction for Russian Empire than for German one.
2. Russians are very good in Northern fighting. Swedes are good too, but Germans not so much. I can see savage guerilla-style forest war between Russian, Swedish and Finnish raiding parties in Southern Finland, with Germans reduced to a role of cannon fodder, being teared apart by Russian plastun raiders.
3. What direction would Finns go? IOTL they somewhat leaned to Germany, hoping that Finland is a "bridge too far" for the Germans and they would have no sensible choice but to grant independence. ITTL Swedes (who're ruled Finland as colony but a century ago in a way that Finnish status within Russian Empire was a marked improvement) are coming (undoubtedly, waving flags of return to former glory, otherwise what's the point for Swedes to enter the bloody mess). And Finns might be just less than thrilled of becoming Swedish province (instead of self-governed territory, with own currency, laws, police, Parliament and custom border with Russian Empire proper) again. And Finns are a force to be reckoned with in their forests.