but I was kinda taken back by the Swedish invasion. By the way isn't John II Casimir part of the Vasa family?

Long story short, the first Vasa to become ruler of Polish–Lithuanian Commonwealth, Sigismund had a very strong claim on Swedish throne. Actually, he was a king of Sweden, even if for a few years, before he was ousted by Charles IX. Despite his attempt to invade Sweden in 1598 was crushed, neither Sigismund nor his sons ever dropped claim on Swedish throne.

And there was also Livonia. After old Livonian Confederation fell in 1560's, the land become contested by Lithuania (and latter Commonwealth), Sweden and Russia. The first open war was fought in 1558–1583. And then another one in 1600–1611 and another one in 1617–1618 and another one in 1621–1625 and yet another one in 1626–1629.

That just Commonwealth vs Sweden front. Swedish had few wars with Muscovy/Russia of they own in this time period, just as Commonwealth did (but frankly, Lithuania was in conflict with Muscovy since early 14th century and Sweden was fighting Novgorod from... even before that. Back story for Deluge is very long).

And then Khmelnytsky Revolt, fallowed by First Northern War happened. Essentially Sweden seen how one of its old adversaries (Commonwealth) is stomped by another old adversary (Russia) and decided to join in. Aside settling old scores and all that Sigismund's claim to Swedish throne thing, Swedish also acted in order to prevent Russia from taking over all or most of Commonwealth- as that seemed possible at the time.

And that's a very crude framework of the setting. Denmark had a large involvement in Livonian conflict, Thirty Years' War war results should be taken in to account, Swedish-Polish alliance was explored diplomatically just before the war and so on and so on, but in shortest possible way to say it: Commonwealth ended up with a short straw in a centuries long multi-sided conflict.