You don't need it always running, if the Steam Client is not launched, it will do so when you launch the game. Neat, huh? However, you do need the client running locally to launch the game at some point. Not sure if you can shut it down after the game comes up. If there is even a reason to.
And, as long as your credentails are stored locally, it should not have to connect to the Steam service on the web. Except to update itself. I think if you tell it you have a Dial Up, it will even prompt you before it tries to update the Client.
Neat... I won't kid anyone, I can't stand Steam. Currently none of my Steam games can be played, as it has lost my password and will not accept it when entered manually.
Spoiler The Password recovery UI really [REALLY!] needs work. When I try to reset my password it asks for my account name, then sends me a code and asks me my secret question. When I type the answers and click "next", it then asks for my new password.
This seems well and good until it becomes clear that Steam has not checked that the secret question answer is correct ~This is VERY counter intuitive, and becomes a compounded problem when you consider that the password dialog-box does not allow the user to paste in their password, but actually REQUIRES them to type it blind ~twice; (One could argue that this is a good thing, but that argument falls apart when Steam refuses to accept the password because the answer to the secret question was wrong). This encourages crap passwords.
I use the GUID generator from Visual Studio for all important passwords (like FTP logins). Steam it seems cannot accept a strong password, so I've had to cripple my password strength to drop down to it's level. Typing this blind ~twice into the password reset box is a pain, especially when it refuses me every time, with the excuse that either I don't know my own "mother's maiden name", or that my new password was invalid.
Right now Steam is useless to me and the commercial products that I have "bought" via Steam are non-functional.
I would check what happens when you try to close close or kill Steam outright during a game ~but I cannot run any at the moment.
There are several ways to recover your Steam login credentials, so I don't think we need to worry about losing our games along with our passwords.

I'm stuck waiting for someone to get back to me ~again. (this is the second time in two months)
For me the only real downside of Steam is that I can't re-sell my game licenses. I really wish they would fix that.
That too. It goes against common fair trade and implies greed by refusing the customer's option to sell something they don't want; requiring others to buy direct instead of from a friend.