My advice would be to have your vet out for a check-up and go over all your concerns with him/her. If you are going to have a horse you are going to need the services of a vet and this would be a good time to start building a relationship. And a check-up now will give a vet a history to work from in the future.
Did the rescue give you a record of vaccinations? This is something else the vet will need to know.
And welcome to the board. Enjoy your new buddy! : )