You don't need to cut up anything, just remove stuff in case you decide to put it back in later. Which you might after you get tired of how loud it is driving around in a metal box with no interior.
Anyways, you could get some inexpensive racing seats like Jamex with tube frame, those are lighter than stock seats and look good and hold you in in the corners. Take out back seats, and the spare tire stuff like you said. Removing the door panels won't hurt your chances of sirviving a crash, but cutting structure out of them will. You could try replacing the rear side windows with Lexan...don't know how much that would cost. If you are looking to keep this a decent daily driver you really need to weigh the cost of losing all your comfort, but if you are going for all-out speed and racing then you can really take this to the extreme. Remove the whole dash and just put in some Autometer Ultralight guages for the necessary info.
Lightweight wheels are a great idea, they improve performance in every way - better braking, better acceleration, more responsive suspension that lasts longer too.