A hot wire leaves no mess. Fuse wire held tight between a wooden frame, rheostat (remember them?) or simply a variable voltage/current power supply (Radio Shack). Turn on the juice and vary the voltage/current until just hot enough. Cuts like a hot knife through butter. It works well with open/closed cell foam too. A bit of overkill for what you want to do.
Alternatively a bread knife does just as good and doesn't leave any residue on the bandsaw. You are also not limited by the throat of the bandsaw.
This particular gate used to rattle about halfway closed for some reason. I used the threaded hole where the lock knob used to be to hold the wood, counterbored the piece of wood and epoxied in a magnet. Seriously, it’s not as elaborate as it sounds. Now the door drags against the magnet and stays in whatever position I happen to put it in.
For the piston action, you can shorten the top of back piece to allow air to escape. Eric recommended that to me a few years ago. I have a cabinet with the same issue, though I never got around to doing anything to address it.