I am wondering if it is legal and/or possible to buy a copy of Final Cut Pro X and then share the disk and the cost with five other people?
No. This would violate the license agreement between you and Apple when you purchase the software.
Second, there is no such thing as an installation DVD for Final Cut Pro X. You would have to purchase it as a download and install and register it through the Mac App Store, and I don't think you would be able to use your single Mac App Store log-in account on six computers owned by six different people who are not family members of yours.
Kinda tricky to answer, because questions of legality are highly dependent on your location. Of course, it's entirely possible that you live in Afghanistan or a small and hitherto unknown spit of land in the middle of a vast ocean without any recognised legal system, or perhaps you roam the skies forever outside the jurisdiction of any sovreign state, occassionaly touching down in airports around the world to wash your underwear in the restrooms, but I'm going to go out on a limb and imagine that you live in, and are therefore bound by the laws of, The U S of A. (God bless)