1. How do I deal with crossing the gate? I don't want to go underground and leave a weak spot.
It's going to depend on how tall your fence is.
If it's tall enough, run the hot wire over the gate to do the other sides.
It doesn't have to run back to the charger.
If it's short, you will have to run a piece of insulated wire in some PVC under the gate.
For the gate itself, you can run a continuous piece of wire around 6 insulators.
What you want will be an "H" shape, but with wire across the top and bottom also.
Then use a piece of
insulated wire to connect to the closest hot wire
on the hinge side with with enough slack to open and close the gate. "Polywire" works well for that sort of thing.
The Nylon kind works much better than the Polypropylene
Another option would be to use "gate hooks" as long as you realize that when they aren't hooked up, you lose power to part of the fence unless you have the underground wire too