No. That's why it is very much legal for me to chop up a tree that I find on my property but not a dog.
Furthermore, farmers can kill your dog (humanly) on his land if it is causing distress to his animals. In law this is a lawful excuse to cause criminal damage. There are all kinds of lawful excuses. For instance, a person (usually a farmer) is free to kill your horse, humanly, if keeping it alive is causing a material financial loss.
Well, that's the whole point.
I thought you were going to say that in the UK, you can't just chop down a tree on your land - there are often legal barriers preventing the felling of listed trees!
I'm not sure what you mean when you say "discounting welfare issues". Sure, if you discount the laws that protect dogs, they're not protected, but The Animal Welfare act (2006) does exist, very much does protect dogs from people "doing what you want with them" and is enforced.
Of course, that protection is not offered at all costs. If a dog attacks a person, or invades farmland and attacks an animal, then the priority of that person/farm animal takes precedence over the dog.
I'm a vegetarian, so dogs are not special to me in that regard. I only mentioned dogs because I was replying to a comment that mentioned dogs specifically.