I'm going to be a student at a public university in Texas, and will live at a fraternity off-campus. But I believe I will still be keeping my California residency, as I'm not "moving" to Texas. I'm going to be buying a car in Texas. Can I register this new car in the state of Texas, despite not having residency nor a permanent address in the state of Texas?
According to driving laws in the US, in Texas:
a non-resident owner of a privately owned passenger car that is not registered in the state in which the person resides and that is not operated for compensation may operate the car in this state for the period in which the car’s license plates are valid.
Texas allows you to register the car in Texas even if you have an out of state title.
However, in your case, because of the amount of taxes, you will probably WANT to register it in Texas. You will likely pay less in state taxes, less in state sales tax and less in licensing fees (probably 1/10th) and less in insurance costs.
Use your address at the dorm. You don't need a Texas drivers license to do it.
Once you get to California you can re-register it there if you want.