If Beto wins the Senate election (which I consider unlikely), there would be a vacancy generated on January 3, 2021 which Governor Abbot would be able to appoint a replacement for until a special election later in 2021.
Beto could choose to serve in the Senate for 17 days between January 3 and January 20.
Could Abbott appoint anyone he wants, or must his nominee be approved by the state parliament?
If he indeed can, must the appointee be a Democrat?
No. That's not the law in Texas. Abbot would definitely appoint a Republican.