How would situation develop?
Depends on a variety of factors, but not many changes immediately. In OTL, following Casimir's agreements with the Lithuanian nobility upon taking the throne of Poland, Lithuania and Poland only shared a monarch and were completely autonomous otherwise - they did not help in each other's wars and only tenuously cooperated together, even less than normal, non-PU allies were expected to cooperate.
With monarch residing in Lithuania, could Grand Duke keep more power?
Absolutely. With every single Grand Duke of Lithuania from 1440 to 1795 "ruling" it from Krakow or Warsaw, only visiting it once every few years and leaving the rest of the domestic and foreign policy matters to the local institutions, Lithuania was effectively an oligarchy run by a few select families of magnates through the Council of Lords. The Grand Duke being constantly present in Vilnius eliminates this development from ever happening.
It's hard to say how Lithuania would develop in such a scenario, given that its entire early modern era was completely defined by oligarchic Council of Lords rule - it could turn into Russian type despotism or it could be a bog standard feudal monarchy, and maybe shift into European absolutism if it lives until that far. Really depends on what monarchs independent Lithuania will get. A series of weaker Grand Dukes would lead it towards Western style rule, a series of tougher dukes or the takeover of an Orthodox dynasty could lead it towards Russian type government.
Could Lithuania survive wars with Moscow hands without extensive Polish help?
Help? What help? When Poland demands a a Union of Lublin just to start giving help, that's the question you've got to raise...
If anything, being in a personal union with Poland (pre-1569) was actually a hindrance to the Lithuanian war effort rather than a boon - and it came down to Alexander Jagiellon. His decision to leave Lithuania to assume the throne of Poland in the middle of a war with Moscow, and, just to make it worse,
take the duchy's treasury with him, was a death knell for Lithuania in the 1500-1502 war with Moscow - without a monarch and without a treasury, Lithuania suffered its worst defeat in its history up until that point, losing a good chunk of their territory, and they never recovered from that loss, leading to the Union of Lublin a few generations later.
Without Lithuanian grand dukes leaving the country for self-serving magnates, practically decapitating it as a result, it would be much more competent in domestic and in military affairs, so it's likely that the Muscovite-Lithuanian Wars would be a much closer ordeal than they were in OTL. Of course, sooner or later, Moscow would gain the advantage - Lithuania being raided by Crimean Tatars + Russia can easily expand east while Lithuania is bottled + a naturally higher population + higher centralization with Oprichnina and similar reforms = a bad time for Lithuanians once these advantages start accumulating. However, if something similar to the Time of Troubles still happens, and Sweden still rises to become a rival to Russia, then I could definitely see Lithuania survive to the 17th century and onward.