Mostly it depends on the contract that is signed when you started renting. It is a customary practice. And if its in the contract you may have to pay the 30days your not there if you leave suddenly.

Agreed, 30 day notice works both ways. Renter must tell you he is leaving in 30 days or you tell him he must leave in 30 days, pretty SOP

Yes
Texas Termination of Lease Law - Lease - Landlord Tenant

And it also depends if that now month to month was previously a lease that required a longer notice, they may still be bound to the longer notice.

OP did not give enough info as to exact terms and type of rental