from p66 of the current Texas outdoor manual, see TPWD web page for downloading or ordering a free copy, it tells. Its the annual update on hunting and fishing regulations in Texas.
"Game animals and game birds may be hunted with any legal firearm, EXCEPT:
* white-tailed deer, mule deer, desert bighorn sheep, and pronghorn antelope may NOT be hunted with rimfire ammunition of any caliber.
* shotguns are the only legal firearm that may be used to hunt turkey during the spring eastern turkey season.
* pellet guns and airguns are not legal.
* fully automatic rifles are not legal.
* firearms equipted with silencers and sound-suppressing devices are not legal.
* migratory game birds may be hunted only with a shotgun
Most of the turkeys here are Rio Grande turkeys, and rifle is legal on them. You have to have landowner permission too. Some places are archery only, shotgun only, just check where you're hunting. Some people don't like rifles in the spring because calling them up is the sport and because the rifle can destroy so much meat.
Yes, 270 is probably the most popular round around here because we mostly hunt whitetail. The blackbuck are smaller and likely a longer shot, the sika are asian elk, but small (like a whitetail in size) and stalky compared to red deer or American elk, the fallow are like a whitetail alos. It sounds great, going after all of that stuff, wow! Take a camera too.
You can use anything for those critters as you would for white/blacktails. I'd say anything from a .243 on up. I've used a .243, and a 6.5X55 Swedish Mauser.
It takes a big blackbuck to weigh 100 lbs. Most Sikas are <150lbs (except the Dybowski subspecies which is up to 100 lbs heavier). Most fallows are the same size as the sika deer, though there is a strain of Fallow deer called a Mesopotamian (this subspecies is usually raised for meat and doesn't usually carry big antlers like the European)that can be double the size of an ordinary European fallow. As ever it is the rifleman, not necessarily the rifle that is the most important.