Does the BLOODLINES game support 360 controllers
Nope. Bloodlines 1 didn't support controllers out-the-box in the old days.
If you want to play this with a controller, I would try and grab a 3rd party app of some kind, which could force controller support for the 360 gamepad.
For 3rd party apps to use a controller with - think like say Joy-To-Key, last freeware version of XPadder, paid version of Controller Companion
, etc etc.
There's profiles for Bloodlines on Controller companion - https://steamcommunity.com/workshop/browse/?appid=367670&searchtext=vampire&childpublishedfileid=0&browsesort=trend&section=readytouseitems
Never tried it, can't speak on it. I do have Controller Companion - but have Bloodlines from retail (on discs!) and on GOG. Never tried Bloodlines with a gamepad, as....I'm used to it with KB/mouse.
Never tried BL via Steam, but...if you use Steam, what about using the game's built-in controller configuration stuff?