You will likely have a very difficult time recovering if you decide to file suit. Typically, in order to recover for a dog bite, you have to show that the owner either violated a leash law (does not appear to be the case here) or that the owner knew his/her dog had dangerous propensities, but negligently allowed the dog in contact with an innocent bystander. It used to be called the "one free bite rule" - a dog owner is not on notice until their dog bites someone the first time. Once on notice of the dog's dangerous tendencies, the owner has a duty to keep their dog under control and keep him from biting someone else. I suspect you will have a hard time proving the other dog owner was negligent. If you can't prove negligence in this type of case, you can't recover. If you injuries are serious, you should talk to a lawyer in your area to explore your rights. Good luck!
Unless you have serious injuries, you should be able to handle this claim by yourself in a justice of the peace court. You should be able to go to your county's website and pull the necessary forms to file the claim yourself, In the alternative, you may have to go to the courthouse to obtain the documents. You can complete them while you are there and file them with the clerk's office.
You can sue. You should contact a lawyer soon.
This answer is given without sufficient information about the matter and therefore cannot be relied upon without further consultation. Also, this answer does not create any lawyer-client relationship.
It depends on the circumstances - was the other dog off leash? Did you let your dog get too close? How did you end up getting bitten? Was a report filed with the police and/or animal control?
You should send a letter to the other dog's owner, with a copy of your medical bill, and ask for payment within X days. If they refuse to pay, then you can take them to small claims court. However, keep in mind that even if you win a judgment (and there is no guarantee you will), collecting a judgment is an entirely other matter. The other dog's owner may well be judgment proof (have not assets, file for bankruptcy, etc).
You may want to speak with a local attorney for further guidance.
If we do not have a signed fee agreement I am not your attorney and this is not legal advice.