The DWI comes from "operating" or being able to operate a vehicle, even off the roadway, so the cop can arrest you when you have the keys and other indicia of operating the vehicle. His right to arrest you-or not-can be tested with a motion to suppress, which you should discuss with an experienced DWi ATTORNEY. Although the arrest is probably OK, it might be worth going to trial to see how a jury would see it.
It depends. Were you the only person there when the cop showed up? The prosecution can use circumstantial evidence to prove you were the driver, particularly if you are the registered owner of the car, and if you had the keys, if the seat was adjusted to your height, or if you admitted to the officer that you were driving when the tire blew out.
I recommend that you get a good DUI lawyer.
This answer is my personal opinion, offered for informational purposes only. It is not a legal opinion, nor is it legal advice, nor does it create an attorney-client relationship with anyone reading it.
The DWI statute requires the State of Texas to prove that you were "operating a motor vehicle while intoxicated." So under your fact scenario, you are correct that the State may not be able to put you behind the wheel and prove "operation." However, you need to see the police report first to see if the officer witnessed you driving prior to the blow out or if there is another witness that can do so. You definitely need to hire a local criminal defense attorney and id the officer handed you a notice of intent to suspend your driver's license, request an administrative license revocation hearing if you have not missed that opportunity to do so (there is a 15 day deadline from the date you receive notice of your license suspension). This will help you defend against the DWI charge and hopefully keep your license from being suspended.