2: You may qualify with any non-rimfire semi-auto and be qualified to carry any handgun (including rimfire). If you qualify with a revolver you may only carry revolvers. The specific make/model do not matter & aren't even recorded. (NOTE: you must use factory ammunition, no reloads)
3: If an LEO asks just tell him you've only been in TX a couple of months. He won't call your boss to verify.
4: You may legally carry, open or concealed, without any license on any property under your control. You may also have a handgun concealed in a vehicle or watercraft without a license.
Oh, and welcome to Texas! You picked a great time of year, it isn't 115 degrees in the shade just yet.