I have an SGS with stock Gingerbread (2.3.3). Can I install CWM without Odin?
Will installing CWM also root the phone, or do I have to root it ahead of time?
You need Odin in order to install CWM without root (or conversely, you need root to install CWM without Odin). CWM does not root the phone.
You don't need root for ClockworkMod Recovery, though you do need root for the ROM Manager app.
Edit: You can't install unsigned packages with the "3E" recovery, which was part of some official update packages (most Froyo ones and presumably all Gingerbread ones). This prevents one from flashing a custom ROM, kernel, modem, or recovery through the stock recovery (since they aren't signed by the manufacturer).
The way to get around this is normally to use Odin to flash a custom ROM or kernel with a custom recovery. I'm not sure if flashing a signed update to an older official ROM, with the older recovery, would work; it might say it's older than your current version and fail, or it might just work. I've never been able to try it, and I'm familiar with Odin anyways.