Should I have my dog checked by a vet before trying this sport, to make sure there is no medical reasons that she should not do this?
Absolutely, but this should be done before doing any dog sport (agility, flyball, disc, etc...)
Do dogs that do weight pulling have issues with there joints/back/neck or any other parts of their bodies later in life??
If dogs are healthy and do not have any structural abnormalities then weight pull can be a fantastic sport. The weight pull harnesses are designed to allow the dog to pull safely and gradually working dogs up in the amount of weight they pull to allow their bodies to adapt and get stronger will help to keep things safe. I think there is a huge misconception with weight pull and the fact that it causes injuries. My Lab Earl blew out his CCL and meniscus at the age of 2 and I use weight pull with him now at the age of 7 to build up muscle mass in his rear. Load bearing exercises are suggested for people of all ages to help build and maintain muscle mass and bone density, and in my opinion, it is no different in dogs.
How often are dogs injured in this sport?
Someone else may need to chime in on their experiences, but I've never seen a dog get injured from pulling safely. I have seen lots of dogs get injured doing flyball, disc, agility, bite sports, etc... Again, like any other sport, pulling needs to be done in a controlled fashion and in a safe manner.
Why do dogs (especially pit bulls) like to do this?
Their size to strength ratio is very high compared to many other breeds and they were bred to give it their all and to never quit. That combination makes them naturals for the sport