"Bill had his house painted" means that another person painted Bill's house, but money is not mentioned so we have no idea if the other person was a paid contractor, or maybe a friend. "Bill paid to paint his house" isn't a correctly formatted sentence. Who did bill pay? We could say "Bill paid his brother to paint his house" "Bill paid for painting his house" this sounds like Bill did something wrong and has been fined or suffered a loss after the house was painted.

"Bill had his house painted" is correct. "Bill paid to paint his house" and "Bill paid for painting his house" are formed incorrectly.

You could say "Bill paid to have his house painted" or if he himself painted his house, you would just say "Bill painted his house"

I think "Bill" is the name of someone, so the sentence is like "someone had something done",means someone did something.