Steam should be caching the item in the background (even when the game is not active) but the file isn't written to your disk in the /addons/workshop folder until the game starts up. The game attempts to do this immediately and in the background as the it's running.

If I have another steam game running, is it also downloading? What about playing anon steam game, some people (read me) have slow internet access and it can be a drain not being able to stop any downloads/uploads.

I believe it'll follow the same rules as downloading any other game content in terms of bandwidth and should limit that when you play other games, etc.