CS5 will be fine, although you will want to make sure you have at least 4 gb ram.
Final Cut really requires a dedicated graphics chip to run effectively. You probably will be able to do basic things on the mini, but motion graphics added to the video or any complicated editing will not work very well.
Do you mean Final Cut Studio or Final Cut Pro? FCP still relies on the CPU and does not take the GPU into account, thus FCP will work with a non-Intel IGP. Motion and Color are the applications in FCS that need the GPU for their tasks.
I meant final cut studio, and was referring to doing motion graphics. Sorry for the confusion. Final cut pro is not going to work that great either on the cpu that comes with the mini in my opinion however. Dedicated graphics memory is necessary for doing longer film clips.