Can I play Freespace 2 with a mouse?

Yes, you can.

You can, though it requires a lot of picking it up and putting it down. The first time I played this game was with a trackball and I still can't imagine playing with out one 10 years later.

You suraly can but why to do that when it will be the worst way? Gamepad or joystick are a must for this genre. It is a sim afterall.

Yes, unlike X-Wing Alliance which won't even start up!

This has been discussed, though. However, they simply don't cost that much and immeasurably improve your game experience. For the cost of a Dualshock or XBox Controller you can have a superior flight experience. Then you can reuse it for X-WIng or a military flight sim. You'll wonder how you got along without it.

I'd advise buying one with a Hall sensor to prevent it wearing out in a couple of years. Like this one: http://www.thrustmaster.com/products/t16000m which runs about $40 on Amazon.

Really, take the plunge.