Weird replies, dogs seem to bring out the worst in ppl.
It does sound like you're trying, OP. It's worth bearing in mind that these things can take quite a while & you've several things going on at once that have built-in settling periods.
The cat moved house.
A dog is introduced to the new space.
The dog is a young rescue who's history is unknown & breed vague.
The dog is neutered very young (no judgement but that was bad advice)
Things to remember -
The dog can get the cat, if he wanted to it would be dead already. Doesn't make it OK but it's important.
He's 10 months old - they regress & appear to have forgotten things they already know.
He's a terrier mix of some sort - so very high prey drive & willful. Worth looking for advice on training a Jack Russell with cats. It will be more specific & even if he's not JR the advice will likely be the most useful cos they are belligerent little shits with small things.
This needs time & perseverance!
Absolutely enforce the no-jumping over the baby gate. Get a taller one so it's a new set of rules & he must never be allowed to jump it. Really really work on that. Work on getting the cat to stay still/slow around him when he's gated with high value cat treats. The cat running is absolutely the trigger, some dogs just cannot resist & you're in a loop of that happening.
Have you got a floor to ceiling car tree with a bed at the top? If not, get one.
I'll say it again - this needs more time. You're expecting a lot to happen in a few months that has a lot of change in it & you can't brush it off as 'just natural behaviour' & leave it at that. He hasn't killed the cat so absolutely can be trained out of it. As you say - this is new behaviour.
It must be driving you both mad & be upsetting for your boyf, really traumatic for the cat & stressful for the dog.