Hi and welcome to the Forum.
First I'd recommend you read our Northern Ireland Guide (top left-hand corner is a link to our "Discover" section, mouse over that and you get a drop down menu - it's in there).
To deal with the visa aspect; as US citizens you do need a visa to live and work in the UK. if your husband can get a job that meets the criteria of the shortage occupation visa (link), then none of you should have any problem getting into the UK; but he does need to find the job and get sponsored by his new employer first, you can't travel, then look for work.
The only thing of note is that all work-related visas are by definition temporary and you have no guarantee at the end of the contract to remain in the UK; that said, you will have the opportunity to apply for settled status, so remain; it just means you have to behave yourselves.
My comments/opinions:
Schooling is difficult; you can home school, but I suspect your eldest will then have no recognisable qualifications; not a good place to be for a young kid looking for work and pretty much means he'll be flipping burgers or filling shelves in a supermarket. That said, unemployment as a whole is very low in the UK, so he should get some kind of work.
The UK does not recognise "emotional support animals"; they'll be treated like every other cat. That said, I can't see any problems providing you have the correct documentation and your cats are microchipped. I just checked and can't see any direct flights from the US to N Ireland, so you'll be landing on the UK mainland first, then travelling onto N Ireland; pets in cabins are not allowed on flights into the UK, so they will be travelling as cargo; I'd advise you speak to your airline well in advance as my experience is that it can get a bit emotional at check-in.
Good that you like awful weather because Northern Ireland has lots of it.
You may find the Numbeo website (link) useful for all sorts of social information (costs/crime/healthcare etc).
If you have any further specific questions, please come back to us.
Hope this helps.
Cynic
Expat Team