When I load up the game my ship spins down and to the left non stop. No matter what button I check or uncheck in the option menu I can't seem to make it so I can play the game. Anyone else have this problem?

I've not seen it but it sounds like a joystick calibration issue. I'd suggest giving it a recalibrate and see if that helps

Or simply disconnecting it to see if the problem goes away.

But there's no joystick to calibrate. I've gotten it to work on my lap top that I use at work. But on my computer at home I get the spins. I don't know if it's my mouse, or my keyboard or what.

Ahhh that DOES rather lower the odds of it being a stick calibration issue. It might be a good idea to erase all the key bindings (you can reset them to default later) and then start the mission up. You won't be able to do anything but if the view moves, it should eliminate the keyboard as the problem.
It might be a toggle key thing, see if capslock, scroll lock or num lock is bound to anything

The same happens to me. When I cleared all of the bindings the first time, it stopped spinning. But when I started to add the bindings one by one to see which one was the cause of it, it started to spin again...even when I cleared all the bindings again.

I noticed that the problem had to do with the "turn (yaw) axis" and "pitch axis" bindings. If you clear those the camera will stop spinning. The downside is that you won't be able to play with a mouse anymore. Does someone have a possible workaround ?

If this happens with FreeSpace 2, try FreeSpace Open. Installation instructions: http://www.hard-light.net/forums/index.php/topic,47640.0.html

Does anybody have a solution to this yet? I hate to bring up old topics, but I'm having this problem, too.

Unfortunately I can't help you fix the vanilla version, but you can try installing Freespace Open, the open source port for Freespace 2.

seems this is another bad version from GOG. Same problem here.