I’m not sure about weasels and climbing but after seeing my little chipmunk climb a huge tree up 25 - 30 feet I’m thinking the weasels might if they were hungry enough. But likely not. LOL! I don’t think a Fisher would climb too high either. The 2 x 4 wire will allow small birds to get into the run. Not predators but could carry disease and steal food. It’s your call.They’ve been destroying mine too!! And were stealing my chicken feed before I moved it inside. Naughty little things! Lol we are overrun with them currently.
I think the welded wire would be fine. Theoretically it shouldn’t be and they can fit through it but I’ve had mine in a chain link run with a welded wire top for years.
I know that’s just anecdotal and that I’ve been extremely lucky but we are pretty wooded and have all sorts of predators and have been fine so in a suburban backyard that close to the city I think you’d probably be fine with welded wire. Especially if you plan on covering part of it and the welded wire would only be on a little bit of it. Is your yard fenced? If so that adds a little bit of protection too. Not a ton, predators can still get through it, but it adds an extra barrier. Ours isn’t and we have had lots of attempts by raccoons and foxes to break in but none have been successful. But I do still plan on putting HC around the rest of the run.