Thanks for the photo
@ChickensComeHome2Roost that's what I was talking about except I use screws to tie the two pieces of wood together instead of nuts and bolts. Either way works. I used it on 2x4 welded wire and it tremendously stiffened it. It was not flapping but was very rigid, but that might depend on what wire you use. You can tie it to the bottom frame if you want to but I did not find that necessary. With yours it could help. Each of these builds is going to be different, you need to adjust it to your situation.
If the opening is big enough it can.
All you are trying to do is to build a pop door between the two. If you want to use that existing gate for that you can. I'd consider leaving that gate as is as a way to gain access to the existing section, even if you have to crawl, and put the connecting pop door on the opposite side of the existing run. That would require a closeable pop door on each run section so you can lock them in each section but that might turn out to be easier.