He is kind of correct, as you are not on his lease. You could edit your gf's rental agreement with your name on it temporarily. Download the APP and do it yourself. Or your gf, who is the Possessor of the property, can do it for you as well. Does not need to be the Landlord.
No one can answer that unless you say what your plans are. Travel in an out regularly? Keep funds in UK or move them over? Insurance in Thailand or UK? etc..
No you don't have a 90 day visa, you have a 60 day tourist visa. When you are issued a tourist visa, you must enter before it expires before 90 days. Right now you can extend for 30 days and then also get a COVID extension after that, if needs be.