Using mobile phone hot spot as Internet provider

I would suggest checking the small print for the sim contract (and trying to find other people on the same package who have done it already), quite often "unlimited data" is really "unlimited data*" where the * denotes that there are terms and conditions such as unlimited meaning they won't cut the connection but will throttle it, or will cut it if you exceed what they consider a "fair" amount.

Yes in theory, but as others have said just check the fine print.

If you are with Three it is fine as their unlimited seems to mean unlimited.

I am assuming you live alone, otherwise when you leave the internet will be leaving with you.

Only other issue might be if you intend to have a smart house, e.g. internet enabled doorbell, heating controls etc. as they need to be connected to wifi.