I'm adding a hotwire 6 inches off the ground of the 5 rail wood fence in my backyard to keep the dogs in. How should I go about running the hotwire across the 3 gates? I imagine using metal connectors which make contact when the gates are closed allowing the current to go through. However, I cannot find any contacts made for this purpose. I've seen one for sliding gates, but not swing gates. Thoughts/ideas?

Looks like one option is burying the line under the gates and just escape proofing the gates.

What about the contact points for window alarms? The only problem I see is keeping it level over time.

Install an insulated run under the gate then run a line at the base of the gate from the hinge side. Make sure there is enough wire so it doesn't ground when opening the gate. It doesn't need to connect on the latch side.

Just make sure you get insulated wire rated for 8-10kV or so, otherwise you'll get an arc to ground through the insulation.

If the gate/ fence is already built and you don't want to dig... they do make insulated handles with spring hooks for hot wires gates.
I'd get one of those, and turn a 90* at the base of the gate on the latch side so the hot handle is at/ near the latch. Just have a separate eye/hook on the gate to hang it on when you open the gate.

Are you needing to continue through the gate to continue to the rest of the fence or are you concerned about having the gate portion hot?

If the latter, I can assure you once your dog hits the hotwire and figures out that is what is shocking it, you can lay out a circle in the middle of the yard not connected to anything and drop a T-Bone steak in the middle and it will be safe!

I had a hot wire for my old dog. It was battery powered. After a while, I stopped replacing the batteries, then took the wire down. He wouldn't cross the line where the wire was without my express permission. Never had an issue with him leaving the area he was supposed to be in after installing the hot wire.