Well I'm certainly not an expert on this topic, but I can just pass on what I know from experience. I know 2 FO's of mine who went through this. One lived and paid tax in MN, the other in RI. Their base was ORD. When they went to file for unemployment benefits in their home states, the state said they had to file for those benefits in IL.

UI is based on where you're based. Presently live in TX, based in NY. No income tax withheld for NY, one dollar something taken out for NY UI. NY UI pays more than TX but is more of a pain to deal with.

Not necessarily. Here is the background, I live in Texas, was based in NY, and my airline was headquartered in Kentucky. I could have received unemployment from any of those states but choose Kentucky because they were the easiest to deal with, I could actually speak to a human being if I had a problem.

Yes, it's which state the company (and maybe you) pay in to. Both my times I lived in TX. One company was also TX based, I was based in IN. TX UI. Second time the company was NY based, I was based overseas. They were taking NY UI out of my paycheck. NY UI.

In a nutshell:

Where your company is based is irrelevant unless you live there too.

Where you are based is considered your tax "home" so few, if any, deductions can be made. You do not have to pay state tax in your tax "home" if you have a permanent residence elsewhere.

Where your permanent residence is, is where you pay state tax.

Correct me if I'm wrong, but I think this was the conclusion we came up with here a few years back when this came up.

Best thing you can do is contact a tax professional where you live. It gets really confusing with all the laws etc.

If I have a car registered in a different state than the state I reside in (go back to visit often often, have a beater car I keep there) does that create any tax or resident issues?

Taxes/commuting
 

My husband (a pilot for Hawaiian Air) has a home and lives in Hawaii. He lived in Florida for school some 25 years ago and uses his old roommates address as his domicile; has never paid state Hawaii tax despite residing here for 25 years. Is this legal? Am
I going to be in trouble for knowing this and not reporting it? I've read horror stories! He has always told me how smart he is to do it this way and I assumed it was legal!

You can google innocent spouse relief
Here are some IRS links about innocent spouse relief

https://www.irs.gov/taxtopics/tc205.html
https://www.irs.gov/pub/irs-pdf/f8857.pdf
https://www.irs.gov/publications/p971/index.html

Now that you know or think you know - You probably need to talk to your husband and a tax professional, as you probably can't claim innocent spouse relief after this point in time.

If Hawaii catches him it's likely they will go after him for criminal tax evasion. He probably should fess up and cut a deal with Hawaii. It would be hard to call this a gray area or tax mistake that would fall under civil rules. He is claiming a fraudulent residence for income tax evasion purposes.
The out would be if he does infact spend significant time in Florida for other than work and can document that time. If however he spends more than 180 days a year in Hawaii it's irrelevant. I would go see a tax attorney with your husband. You do have possible liability if you signed joint returns however I would be shocked if they attempted any criminal action as a spouse. Your husband however could face jail time if he gets caught. How likely getting caught is something he has to decide but I would not sign another joint return unless you claim Hawaii.
The key is you need to consult with a tax attorney not a accountant. He will know how aggressive Hawaii is with this type of situation and can offer the least risk solution.