They are GIRLS first of all, were they boys they would have been treated Ivan VI style - but underage girls are NOT contenders to the throne, bastards or not - except for some wild Menshikov scheme, they are going to get zero support from anyone. So their arrest etc makes no sense and it would be much easier to brought them up in customary noble fashion with emphasis on church education, probably entrust them to the care of Natalia Alexeevna, as Alexei himself was/
Of course, it can be argued that the whole family of Ivan VI had been imprisoned, including females, but, OTOH, their status was completely different: they were legitimate children (stories about Peter’s early marriage to Catherine are rather shaky,why would he marry her the second time) and belonged to the influential ruling family, which could make them useful tools in the attempts to overthrow Elizabeth. OTOH, Peter’s illegitimate daughters from a washerwoman has been nobodies (in OTL the fact that Catherine was a crowned empress who reigned on her own did a lot to strengthen Elizabeth position (she was officially recognized as Tsesarevna).
Now, getting back to Alexei, he inherits a low intensity war in which Russia is on a winning side and the burden of military activities is shifting to the new players, mostly Denmark and Prussia but both Saxony and Hanover also have their interests. In OTL, while dealing with Stenbock required Russian intervention, the Russians had been asked to keep out of the siege os Stralsund and Peter already had to give Stettin to Prussia. In other words, his policies in the region resulted in what later became a pattern: Russia fighting for somebody else’s interests. The only remaining meaningful theater was Finland, which had been mostly occupied before Charles returned (and which was returned to Sweden by peace treaty).
We may assume that (just to make things different from OTL) that Alexei is scaling down Russian activities in Pomerania immediately after Stenbock (the only reasonably serious Swedish force at that time) is dealt with. Russian troops are leaving Germany but probably retain at least some military presence in the PLC helping August to deal with the pro-Stanislav confederates. Most probably, not-interference into the German affairs would help to maintain the friendly relations with the Brits and Dutch (in OTL at least the Brits ended up siding with the Swedes even if not doing too much).
OTOH, more troops could be sent to Finland and the same goes for the intensified seaborn operations conducted by the galley fleet (IIRC, their main goal was to cripple Swedish ironworks along the coast).
With the exception of Menshikov (not sure that he would be compatible with Alexei), Alexei is surrounded with his father’s people and his highest-ranking real associate, Alexander Kikin, was a ....er... quite “progressive” figure judging by his career and his palace in St Petersburg
IMO, a reasonable thing to assume would be approximately the same domestic policies with a lesser experimenting in the area of economy. In OTL,probably the most successful area in Russian economy was a textile industry to which, after few Peter’s disastrous experiments, government almost stopped paying attention. Mining and metallurgy, had been an area of an intensive profiteering by the high-ranking court personages and, while for a while Russia became one of the greatest iron producers, most of that production had been sold abroad (more profit in gold then bothering with producing things and selling them in Russia) and Russia had to buy the finished products. Part of the schema was declaring the found resources a state property and then giving them away to the high-ranking court personages who would be interested in the immediate profit rather than in arranging for long-term proper exploitation. Not sure that Alexei would have brains, knowledge and character to change system dramatically but even the lesser interference would be beneficial.
Now, about the domestic administration. The Boyar Duma while not officially disbanded is pretty much non-existent and replasec by the Collegium (IIRC), which is pretty much a council of the ministers. Would Alexei create Senate or its equivalent?
Any other ideas? (I need to think about them)