This video is of one guy dubbed 4 times. Is it possible to do this with the "Movie Edit Pro 2016 Plus" program? If so, what features/steps would I need to use?
MEP handles audio nicely. There is a very good mixer and there are object and track audio FX, possibility to add in and use third party VST's, object cleaning/restoration tools light, and an external program that most users don't know about, Music Editor 3, which allows you to do more cleaning/restoration and mastering.
You can also automatically align audio tracks recorded with different devices/cameras.
Make sure that you get at least the Plus version as the basic version will likely not do what you want. The Premium version is just the Plus version with some third party add-ons that are usually quite useful.