1 - In the past, they have used New Japan Pro Wrestling matches for their Chris Benoit DVD and once on Raw. Do they own that footage?
No, WWE at the time licensed the footage from New Japan and their TV rights holders, going through the Inoki family to acquire those rights. They do not own any of the New Japan library.
2 - How come WWE didn't buy the MLW rights when Court Bauer worked there? It seemed to me like they would want to get that material since it aired nationally via DirecTV?
I once asked Bauer about that and he told me they showed interest but at the time, he claimed he didn't have any ownership of the footage as he had sold them to an overseas company called WGO. At the time, no one ever seemed to find any real evidence of them existing or using the footage. Since leaving WWE, Bauer has begun using the MLW name online again and claims rights to the footage, so if he wanted to sell and they wanted to acquire it, WWE could. Bauer working for them would have nothing to do with it.
3 - From time to time, I see WWE use some Global Wrestling Federation material, usually Booker T matches, online. Do they own the ESPN GWF tapes?
I don't believe so, but I don't know 100% either way. When I've asked people within the company about the use of those tapes, no one seemed to know what the situation was. I was always under the impression that the Savoldi family had acquired the rights to use that material, but it's possible that Grey Pierson, who backed the promotion still has the tapes. It could be that WWE quietly came to terms with them (which does happen, such as when they acquired the Rick Bassman UPW tapes) or that some footage was included in the Kevin Von Erich/World Class tape library when those Texas tapes were acquired. No one seems to have a clear answer.
4 - From time to time, WWE has used Memphis footage of Jerry Lawler and once used a music video of the Fabulous Ones to mock them on "Are You Serious?" Is it possible they now own that library?
It is possible, but not likely. My theory is that they acquired some footage from Jerry Lawler (one of the matches they have used on WWE 24/7 is Lawler vs. Terry Funk in the Empty Arena match) or that there was some misc. Memphis footage mixed in one of the other acquired libraries, most likely Florida. It wasn't unusual in the territory era for one promotion to use footage from another company when a talent was coming in, so that could be the origin right there. The actual Memphis library is at best, fragmented between the ownership of WMC-TV, Jerry Lawler and Jerry Jarrett, and most likely, no longer in existence besides fan videotapes.
5 - Last question: Where are the Continental Wrestling Federation tapes, especially the Eddie Gilbert era? That TV was so good, I can't imagine that WWE doesn't want it. It's the early days of Paul Heyman, Sid Vicious, etc.
Those tapes are likely owned by David Woods or his estate, if they even still exist. The fact that no one has tried to bring them forward into any sort of DVD deal, sale, etc. makes me think that whatever tapes are out there are dispersed to the winds. That was an incredible promotion. If there is any material out there from prior to Woods owning the company, my guess is that whatever material would be owned by the Fuller family, but again, my gut is there isn't much out there, if anything. That's a shame but that was the reality of TV back then - and why there's not much footage from Southeastern Championship Wrestling, Portland Wrestling, California, etc. out there either. It was all junked.