I work in insurance, specifically in Umbrella.
Does umbrella insurance cover everything, like a blanket, even further than something like homeowner's would cover? Or would the umbrella not be helpful if I was facing a situation where the insurance company said their homeowner's coverage didn't apply due to XYZ.
Typically, a personal Umbrella policy only extends the limits you have on your homeowners policy. For example, if you have a $500,000 limit for liability on your homeowners policy and bought a $1M umbrella policy, you now have coverage for claims up to $1.5M. Plus, defense expenses aren't usually included in that, so that's an additional benefit for both policies.
When you're considering your exposed assets, remember that all tax-exempt retirement accounts are protected under federal law.
I don't typically recommend umbrella policies, I don't buy one myself, but I think you should get one for a limit that would make you feel comfortable. You're worried about something that has an absolutely minuscule chance of happening, and insurance can relieve that worry. So what if umbrella insurance doesn't make financial sense from a policyholder perspective; you're buying peace of mind.
One thing that sticks out to me is that your homeowners policy probably doesn't include your dad as an insured. So, if he gets sued in his own name, is there coverage? Ask your insurance agent; there might be a way to add him. If not, I'd highly recommend buying him a rental insurance policy. (Don't ask your lawyer this question; insurance coverage is a fairly specific area of law that a family or elder care attorney is probably not familiar with, so you don't want to pay for worse advice. Plus, if your agent gives you the wrong advice, you can make a claim against their E&O insurance.)
I just read my policy and I don't see anything that shows coverage would be denied for a nuisance law violation or anything like that. In fact, the text regarding personal liability coverages, exclusions, and conditions is actually pretty straight forward.
This is pretty standard, you're not likely to see something excluded that would be completely out of your control. So they might exclude damages from you owning a dangerous breed of dog, but they wouldn't exclude damage someone else's dog caused to your property. Does that make sense?
Also, while it's true that some states will allow an insurance company to cancel a policy if they find out that there's some undesirable part of the risk, this is almost never on a retroactive basis. When it is, it's typically limited to the first 30 or 60 days of an insurance company writing the policy. To avoid this, I'd suggest sticking with the same carrier for many years. That way, if something happens, you can say "You've been writing my policy for 10 years, you never asked me for any updates and had plenty of time to come out and inspect [your policy almost certainly allows them to inspect the property] if you wanted. I'll calling the NC attorney general." And then you call your AG, say the magic words "bad faith insurance company" and you'll immediately have some big dogs on your side.