The area is very flat. If "mountain biking" requires elevation changes, that isn't going to happen.
The National Park Service operates the Padre Island National Seashore. All beaches in Texas are public, but the NPS owns the entire width of the barrier island within the National Seashore. There isn't a road through this, but you can access it at Corpus Christi. At both Corpus Christi and Brownsville you can get as much touristy beach stuff as you want.
The Aransas National Wildlife Refuge has some cool birds.
I have heard good things about kayaking in Matagorda Bay (if you are skilled) and camping on the barrier islands shielding it.
If you are a big nerd like me, you'll enjoy being in the easternmost county with more square miles than people - Kenedy County between Brownsville and Corpus Christi. That gives you an idea how barren that stretch is.
You'll be going pretty close to Houston, and if the weather's good I hear kayaking down Buffalo Bayou through the city is pretty fun. Somebody did die doing this during a rainstorm recently, but he probably should have seen that coming. If you're in Houston,
reserve the free boat tour of the Houston Ship Channel. The industrial and shipping facilities are really impressive.
Texas is not Florida and winter can happen there even pretty far south. Last time I was in Corpus Christi it was February and it drizzled in the 50s the entire time. Brownsville is a lot further south, though, so you're likely to be better off doing your warm beach stuff there.