A home equity loan is exactly what people usually do. Depending on your credit rating you might be able to get a line of credit on the home which works something like an "on demand" loan.

But this is very important, if you get a loan secured by equity in your house, you can lose your house if you don't pay it back. So check the loan terms carefully. Given that professional painters will generally paint the place for much less than $10,000 you should be able to do this. Also you should be able to talk to the city and show them proof that you're in the process of getting the house painted and they should be willing to give you extra time.

Welcome to adulthood, parts of it really suck.