Will my iMac stay on if I use it's display in target display mode for a Mac mini?
Yes you can switch painlessly from one to the other using ⌘+F2.

Here's the Apple knowledge base article about that, you may find it interesting in order to understand exactly how the target display mode works.

However, I'm not sure you can connect via Remote Desktop without having a physical screen connected (except if you already upgraded to Lion, the "virtual screen" functionality should do the trick). Unfortunately, I can't try this at home as I don't have any Macs without internal display.