Is it legal to install one copy of Final Cut Pro X on six different Macs owned by six different people? 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.