Here's an old one: https://chrome.google.com/webstore/detail/command-conquer-tiberium/jgaeopgjojikeoiidmfaejkifhgjoooe Command & Conquer Tiberium Alliances. Plenty more games here: https://chrome.google.com/webstore/category/app/3-games
Yeah, the Google store has a bunch of good free games and even trials like Bastion. I have noticed that some flash games don't work in Linux because Adobe isn't updating flash anymore for Linux. But HTML5 games work fine
Yes: Battlefield Play4Free Rush Team CS Portable Contract Killer Red Crucible 2 Apocalypse City BeGone: Last Stand HD BeGone: Guerra Counter-Strike Online Verdun Counter Strike Russias Army Combat 3 Sniper Team and even more... Don't know if they will run on Linux or not. Some use the Unity engine, but there is no Linux Unity Web plugin available yet.