In Texas, can I kick out a paying roommate if they're not on any kind of lease? They just pay cash?

Depends on what you mean by "kick out". If you mean throw them out overnight, then the answer is no. If you mean giving them proper notice to leave, then the answer is yes.

As they aren't under any contract, you just need to follow your state laws. In Texas, you should give 30 day written notice (assuming he pays monthly). It would take affect on the 1st of the next rental month. That means that if you gave notice today, the 30 days wouldn't start until April 1 giving him until April 30 to be out. (If he pays weekly, you only need give 7 day notice.)

If you two have no contract, than it month to month . So 30 day notice registered letter should do it .

You have to give them proper notice to leave.

They have to be legally evicted.

No you can't