For the claim 'hunters don't come to your house and injure your pet'.
(Many years ago) We asked the local hunt to refrain from coming into our property after one of our mares was injured.
Sadly they did not stop the next time a livery horse was so spooked he damaged his leg so badly that he had to be destroyed.
I speak as someone who hunted until 25 yrs ago. Many domestic pets have been injured killed in the course of a hunt.
In fact I tried to list to an RSPCA article, but it was down the page so a c&p
June/July.. 1998 Caged foxes found on hunt land, North Yorkshire
September Repeated trespass by hunt, Hertfordshire
Pet cat killed by hounds, South Pembrokeshire
October Hounds rampage across country park, Severn Valley
Hounds killed by train, Shropshire
Hounds killed by passenger train, Pembrokeshire
Hunt master fined after crayfish killed by pollution, Gloucestershire
November Hound killed by car, York
Cat killed by hounds, Shropshire
Sheep killed by hounds, Dorset
Hounds killed by train, Cheshire
Hounds killed by truck, Derbyshire
Decomposing carcasses cause pollution, Monmouthshire
Anti-hunt farmer finds dead fox hanging, Gloucestershire
Hounds overrun animal sanctuary, Lincolnshire
December Hunt trespass farm, Dorset
Pet cat killed by hounds, Llanbister,Wales
Hounds killed in road accident, Co.Durham
Pet cat killed by hounds, North Yorkshire
Hunt horse injured in collision with car, West Sussex
Hunt hounds run wild in crematorium, Warwickshire
Hounds killed on dual carriageway, South Devon
January...1999 Trespass by hunt, West Sussex
Extensive hunt trespass,Kent
Roe deer kicked in the head by huntsman, Devon
Hounds tear through garden and flock of sheep, Shropshire
Fox mistreated by hunt terriermen, Gloucestershire
Artificial earth found on land belonging to hunt, North Yorkshire
Huntsman stamps on fox, New Forest
February Churchills graveyard trampled by hunt, Oxfordshire
Vet finds fox in severe shock, West Sussex
March Cat killed by hounds, Yorkshire
Stag cornered by hounds in school playground, North Devon
April Stag mauled by hounds, Devon
I am neither pro nor anti, but to claim the hunters do not harm to pet animal is ridiculous.