I’m not sure where you live, what kind of snow or ice loads or wind loads you may be looking at. That could be important.
The reason studs are normally 16” or 24” is that paneling or plywood is usually 4 feet wide. You can nail or screw both sides of that to studs at 16” or 24” centers. Since you are not attaching paneling that’s not a consideration. However you have other considerations.
I’m a retired civil engineer but I’m not sure how I can explain a basic tenet of structural design. I’ll try. As long as your studs stay straight, they can carry a lot of roof weight. But if they start to bend their capacity to support weight quickly goes way down. The longer those studs are the worse this effect becomes. When you attach paneling or plywood to those 16” or 24” studs, you stop them from bending. You are not doing that. You are using wire.
When you attach wire, provided it is stretched fairly tightly, you get some benefit to keep those studs from bending, as long as the wire goes on across. At corners or at your door frame, the wire does not go on across, it stops. You are not getting much benefit from the wire there. This is why your snow, ice, and wind loads as well as the weight of your roof become important. Your local building codes should give you a weight to use for design. Numbers in the range of 20 to 40 pounds per square foot are typical. Up north they could be quite a bit higher. On an 8’ x 16’ run this amounts to 1-1/4 ton for 20 pounds and 2-1/2 tons for 40. That’s going to be spread out over all your studs and verticals so it’s not huge, but it can cause problems.
You may have another problem. I don’t know what kind of door you are putting on there or what the frame looks like. Unless the stud or whatever your vertical member is has support, the weight of the door can cause it to sag. Your door may start to jam or not close. You need to keep whatever you use to support that door straight.
I don’t get hurricanes here but I do occasionally get 70 mile an hour straight line winds. A shed roof came off in that. The people that built it used fairly short smooth nails to hold the roof on. It did not work. I mention that to show that wind loads can be pretty strong on a roof. Don’t ignore them. Those loads can be up or down, depending on wind direction.
So what can you do? Even if I knew where you are and what your building code design numbers were I’m not going to do any design. I don’t do that for my own. I don’t know how high your walls are. I’m guessing one is 8 feet and the other is either more or less so you get enough slope so water runs off. If you build it too flat, water won’t run off but will either leak through or sit up there and cause your wood to rot. You need a sloped roof.
Unless you live in a really severe climate your 40” spacing should be OK. On your corners I’d use two 2x4’s, nailed or better screwed together to form am “L” shape. Use enough screws or nails so they are really attached together. That’s a lot stronger than just putting them back to back. That should be enough.
Not knowing what your door framing will look like or how heavy your door will be, I’d suggest beefing it up and running a horizontal 2x4 from the middle of that upright over to the next stud. That will make it about 4 times as strong against roof load, but more importantly help keep that from bending with the weight of the door. I’m a lot more worried about the weight of the door causing the support to bend, leading to your door jamming, than snow or ice or wind causing the roof to just crash down.
If you want the wire you use to sort of disappear, paint it dark green or black. A roller is probably the easiest thing to do but you will still waste a lot of paint. I did that (dark green) just to use up some leftover paint in a few areas. It makes it a lot easier to see inside the run from a distance.
Am I overthinking this? Absolutely yes. But beefing up your corners by using two 2x4’s in an “L” shape and beefing up your supports for the door so it doesn’t jam on you take a lot of worry out of this.