Something messed up my pc which ultimately led to my computer doc having to reset everything. Among the items that he copied and successfully reloaded were several games that I was in the midst of. I had included them for their titles. Since downloaded games are not part of his expertise, he cautioned me about the safety of opening the applications of those games.

Most of the games are from BFG. Redownloading them from my account on their website means I have to start over, which I really don't want to have to do. What do you think? Can I open the copied BFG applications so I can continue the game where I left off??

I don't see why not. You don't say what "messed up" your computer. If it was some kind of virus, it's not impossible things like games were infected, but it's not likely. Some of them may or may not work though. You might need to reactivate. You won't know until you try.