What kind of paperwork I need to have if the homeowner offer free room rental to trade my service of doing housework per day ?

I think a lease would be appropriate, but the standard forms won't apply. You'll want something that talks about exactly what you are going to be required to do, and what rights you will receive. Alternatively, the transaction could be treated as a contract for your services, paid by providing you a room, but the same issues apply.

I think that a standard Texas residential lease could work so long as you and the homeowner are clear in the lease specifically just what housework is expected to be performed in lieu of money and how often. 'Housework' is too vague and can cover a myriad of things. Regarding the room, if you are leasing it, you will continue to have a reasonable expectation of privacy, however, you could address that with the landlord, such as being allowed to put in a doorknob that locks (you could offer to pay for it).