No monthly fee. just the one-time price when of buying the game.
The answer to each of your other questions is YES.
STEAM is a game management/authentication tool for games made by valve.
You can download it now or as part of installing HL2, once you get it.
Steam with your internet connection is required to authenticate/register the game to an email address of your choosing. After that, you can play offline for single player, but if you play on any if thousands of (free) servers on the internet, you will connect via Steam. It is only as anonymous as your email address.
of course an infinite number of email addresses are available for anyone via hotmal/yahoo, etc.