Vincent, there may be some "disadvantages" to starting as an adult, but remember that there are lots of advantages as well. To start with, you are playing piano because it was your own choice, and you have clear goals. These two elements alone give adults a huge advantage. Next, adults are very smart and can plan and be systematic, use these abilities to your advantage. And finally, when you are able to play these pieces, you have your whole life up to this point to draw on for expression and feeling, so you will be able to play wth an added element that many children are just to young to have.
Now, if you have a teacher, that should help speed things up a great deal. Does your teacher know that Fur Elise and the Moonlight are two of your "goal pieces?" If not, make sure you tell her. With your current struggles, get her to show you different practice tricks, ask her to share with you every practice technique she knows, so that you're not just "muscling" through and playing it over and over. You want to be clever about how you approach problems. If it's trills, she should be able to show you some exercises that will help you improve them. For hand coordination and speed, practice HS and HT, very very slow and then faster, back to slow and then faster than before.
Regarding physical and mental disadvantages... Don't look at it in such a negative light, but think of yourself as an athlete, and do some "cross-training." Stretch before and after piano practice, walk and keep your circulation up (good blood flow to the brain is always good!) Take your health very seriously, make sure you get enough sleep and eat well. And, like the top athletes, visualize your pianistic success. Look over the score for the Minuet and visualize yourself playing those trills, give yourself a pep talk before you go to bed, and dream of pianistic progress!