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