Your plan doesn’t sound that bad but I suggest the first good fence you build is the outside fence. Use 2x4 welded wire, maybe with rabbit wire along the bottom and electrify the fence. Call Premiere Fencing and talk to them about that. They are pretty helpful, that’s their business. Talk to them about how to wire your gates too. You’re right, gates can be the weak point. You might consider putting in a sill, say a 4”x6” timber to butt the gate against at the bottom.
Once you have the outside secured, all you need to do is keep your chickens in. Chicken wire can work for that quite well and it is cheap. Just make the moat tall enough so you can walk in there. Unless you have a pretty high chicken density they will not eat all the green stuff in there. They’ll eat the stuff they like and let the other stuff grow. You’ll probably need to get in there to mow or weed-eat two or three times a year.
Deer can jump high or they can jump wide. They don’t do both. So a reasonably high fence you can walk under with another one inside makes a good deer fence.