I believe I’ve read that land owners are allowed to shoot predators/varmints without a license. If that is correct, is the land owner only allowed to shoot the animals if they are actively chasing a calf, or hogs tearing up planted crops, or they are allowed to shoot them anywhere on the property if seen, or even hunt over a feeder?

I understand that there's very little limit on protecting your property be it livestock fence crops whatever. Pest animals like hog I don't think there's any real limit on hunting them. Something like deer or fur bearing animals? I do recall that yes you can shoot if it's to protect property (eg. Fox killing your chickens) but are not meant to retain any part of the animal. Of course, that's just about impossible to really enforce...

I remember back in early 70's a rancher south of Merkel,Texas had been over run with deer in his wheat fields and ask TPWD to have a extended youth or any one to thin out the deer and they told him no, he told them he would shoot them a let the buzzards eat them.They threw a fit but as long as he didn't touch them it was legal( he had talked to a lawyer) Guess what TPWD let him thin them out with a extended season.

Many years ago, one of my neighbors had some dogs that enjoyed chasing other neighbors cattle. Finally, the dogs didn’t make it home. The owner called the Sheriffs office figuring that someone must have shot the dogs.

Sheriffs deputy checked all the neighbors including me. We talked and he never asked if I had done it but asked if I knew any reason for the dogs not making it home. I asked if he meant the dogs that were chasing calves. He looked at me and said “No further questions. Dogs must have strayed.”.