There are farmers who are anti Fox Hunting, the Hunt will be informed of this and try to keep Hounds off their land and certainly the Field will not trespass.
As for Hounds making good pets the answer is usually an emphatic NO!
The reason for this is that they have such a strong hunting instinct that that is what they will do and travel miles in doing so.
I walked Foxhound pups for years, this means the couple of pups would come to me at around 10 weeks old and I would rear them on the farm, lead training, taking them on walks and teaching them not to chase sheep or cattle or deer. This was fine until they hit e 6 - 9 month age when the pair would go off on adventures on their own. Then they would be returned to the Hunt to learn their trade.
I will add that, as a breed certainly in the U.K. Foxhounds are probably one of the soundest breeds. If they cannot do the job they were bred for then they are euthanased. Breeding is very selective.
As for farm dogs, it depends on what is wanted from the dog, Shepherds will want good working Border Collies, also good on cattle. In Australia a lot of Kelpies or Aus. Cattle dogs are used on bovines.
A shepherd might also go for a guardian dog, one who lives with the animals 24/7 protecting them from predators. The Anatolian Shepherd, Maremarrs or Pyrenians.
GSDs were originally bred to be walking fences for stock, particularly sheep. A flock would be put in an area and the GSD would keep them in that area not allowing them to stay and also protecting them.
Corgis were used as herding dogs and some still have then for bringing in the dairy cows but would not be a lot of use with long distance round ups.
Mastiffs, Rottweilers and the heavy breeds are not so great as they need time to groom before being worked, are easily tired and sadly do not have a long life (generally speaking) much of this is due to the standard being altered for the show ring.
Haven't a clue what a black mouthed cur is.