I would think they'd need the full name, with "Beto" in parentheses inserted after the legal first name as they do in obits. Believe I've seen this before.
I looked at the sample ballot and he is listed as Beto - O’Rourke. Check for yourself - https://webservices.sos.state.tx.us/ballot-cert/report.aspx
No. They are not allowed to have nicknames. That was one of the failings of the election system Mitt Romney faced.
I don’t vote in Texas, but it is my understanding that when Mitt ran, they weren’t allowed there and in several other states, to put his nickname on the ballot. Even in parenthesis.