If you buy a game from a site like green man gaming, does it go in your steam library?

yes...as long as it is steam redeemable. the site should say if it can be activated on steam. Alternatively, if you don't want it in Steam, sites often have their own downloadable application like Capsule etc...

It'll say whether or not it's Steam-redeemable.

If the site provides you with a Steam-Redeemable Key then yes, or you can give the key away to someone else.

Some websites though, such as GMG and Gamefly, use their own service simialr to Steam. If a game on the website says "Steam DRM", then it'll provde you with the key. If not, you'll have to use the site's client. (For instance, the service for some of GMG's games is called Capsule.)

You can also add applications to your local game library for Steam to launch. As an example, I don't have the original GW on Steam but I do have it locally installed. It's in my local library for me to play, should I be so inclined.

only if it says that you also get a steam key

make sure it says it requires a 3rd party client - steam

Once you buy it it will give you a steam key and then you can apply it to steam

DONT assume because its on steam GMGs version is on steam. Example: Borderlands 1 - Their version uses capsule