My first question is: what does "fully trained" mean? Level 2? Prix St. George?
Any horse can be retrained from one discipline to another; that's the least of your worries. You should be able to introduce this horse to jumping unless it has an unsoundness that prevents it from jumping. But my suspicion is that, if a well-trained dressage horse is being GIVEN away, it may indeed have soundness issues -- or not be quite so "well-trained" (i.e., be an ill-mannered lout.) So make sure your instructor checks out this horse thoroughly -- preferably without you there the first time if you think you tend to fall head over heels in love with hopeless cases.
Veterinary, dental care, and farriery do vary widely, so your best bet is to call around and get a general idea of what local practitioners charge. Farm calls from the vet can be as much as $150-200 just for showing up, but this cost can be split if several horse owners use the same vet and have their horses seen on the same day. I'm not sure which vaccines are required in your area: my horse gets Rabies,West Nile, Flu-Rhino, Eastern/Western Encephalitis, and Tetanus, which together cost about $150. (And several of these vaccines require fall boosters, so there-s another farm call and more money for shots.) Coggins is $25. So yeah -- about $300 annually for ROUTINE care. Obviously, you should try to establish a savings account for emergencies with as much as you can afford to put aside. A colic episode 5 years ago set me back $1,500. Most equine dentists in my area charge $125-$150 per visit, but you can have your horse's teeth seen to by the vet when he's there anyway. If your horse requires floatiing only, that might cost only $50-$60. Farriery depends on whether you plan to shoe or keep him barefoot. If, worst case scenario, the horse requires special shoes on all four feet, you're definitely going to spend over $100 every 6-8 weeks; if he's fine barefoot, trimming only might be as cheap as $35 each time.
My last concern is the amount of time you'd have to work for the free board. You're a 16-year-old high-school student. You're in school, you have homework, you have a social life, maybe you do volunteer work or other activities, you have family responsibilities and household chores, maybe even a part-time job -- and you're going to have to WORK for your trainer 4 days a week! How are you going to juggle all that? Even more important, when are you going to find time to RIDE this horse that you'd be slaving away for?
Teenagers do, of course, own horses. Often, they also work for their instructors, compete at relatively high levels, AND maintain honor-roll grades. But they are able to do all that because they have their parents' financial and emotional support. So in addition to checking out the horse and calling practitioners to learn their fees, you'll need to have a frank conversation with your parents to figure out how to organize your life to make this happen.