Texas love executing people. Since 1976 Texas executed 498 people, the state in second place is Virginia with 110. Also, their supreme court does not hear criminal matters, only civil, because it makes the appeals process for executing people faster.
But yeah, the law is a strange thing.
No. Texas simply has separate civil and criminal courts. The Court of Criminal Appeals is the supreme court for criminal cases. The Texas Supreme Court is the supreme court for civil cases. A convict can appeal through the intermediate appellate courts and then the Court of Criminal Appeals (and then perhaps later state-level collateral attack or even federal habeas). The criminal appellate process in Texas has as many steps in it as most states have.
The worst, incidentally, is Oklahoma, at about 1 per 40,000 residents.
Median 1
Texas 20
Ok 25