Yes, you can. But it isn't like using a standard external, you go through the TimeMachine GUI.
Why not partition your backup disk? Then you can have a section for Time Machine (over 2x your internal HDD size), and a section to use as a normal external hard drive. They even show up as two normal different drives for ease of use.
Nah. That's not really useful unless the OP is going to install Mac OS X on the other partition. Once you start using backup, there will be file created in the drive, and everything else will be stored inside. If you want to store something like 50GB of home movies and bunch of other files, you can just use it like a normal external HDD.
Doesn't seem normal to me. As long as a Finder window has the focus, that should be what TimeMachine starts with.
OK, I just did. If Mail has focus ("Mail" in the menu bar) then Time Machine opens on Mail. If Finder has focus then TimeMachine opens on Finder. If programs not supported by time machine have focus then TimeMachine opens on the Desktop in Finder. Nothing unusual there.