You mean is it OK to shoot your neighbor's dog Benji if he saunters over there? Or do you mean wildlife? Not sure what you mean by "stray animals".
First, is discharging a firearm legal in your municipality? In your county? Mine for example, allows NO discharge of firearms within the county limits.
Second, certain animals are illegal to kill ANYWHERE (endangered speciies list), so they would NOT be OK to shoot either on, or off your property.
Likely some animals may only be hunted during a given limited hunting season.
SOOOOO, contact your munipality about their restrictions, and contact the Texas version of the State Department of Natural Resources and inquire exactly what may legally be shot in your state and during what season.