I know a person who started to play seriously at the age of 17 - having dabbled a bit, unseriously, earlier - and now is on h** way to become a professional concert pianist. Anything is possible.
As you write that you are going to Med school I assume you are not planning to be a pianist as a profession. I am sure you can learn the pieces you mention; I have no idea how long this will take you though. Maybe shorter, maybe longer than 10 years from now - it all depends on how much devotion you eventually will be able to put in your piano training.
I'd say, just go with the flow and work. Don't worry so much. If you expect someone to shout "oh, you are a genious!" one day because of your fast progress and extraordinary talent, then skip it. But don't let any wiseguy tell you what is possible and what is not either. The trick is simply to believe in yourself, not to ask for approval, not giving up - just work on, the natural way.
Either you will find, one day, that you can play what you dreamed of playing and even more. Or, you look back with a shrug and say "so - I changed my mind". And that will be perfectly fine too.
It is not about which age you started playing. This isn't olympic gymnastics. Don't listen to the garbage people say about brain plasticity, finger dexterity and other BS, you will find that none of them is an expert anyway. You can learn and develop all your life ... if you believe you can.
It is not, actually, about practicing so-and-so many hours a day, a week, so-and-so many years. What really matters is that you practice in a wise and effective way, humbly listen to your teachers, and that you hang on. If you love piano playing enough, you will get wherever you want to go. So, your only real limitation is your own enthusiasm for piano playing. (Not for money, fame nor pleasing anyone else, but for piano playing.)
You must also be slightly fanatic about going to concerts, listening to recordings, studying literature, talking to other students, watching videos, eating, living, breathing piano music. You have got a teacher and that is good - don't hesitate, when you have advanced, to also find "guest teachers". Every new person you meet will give you new insights, no matter if they are piano super stars or beginners.
For example, if you want to be able to play the Hungarian Rhapsody 2 (I want that too) you can start studying it long before you can really start playing it. You can listen to it until you know every little note of it. You can use it for analyzing, theory studies, whatever. You might very well be able to play little snippets of it. And with this approach you will soon discover what you need to learn and practice. After all, exercises are so much more fun to do if you know exactly WHY you do them. I think one of the most common mistakes piano students do, is that they do tons of exercises without understanding the point with them. They just play their scales and their Hanons or whatever, for hours and hours and years, and all they develop is absentmindness and sometimes an arrogant attitude of "I must be such a good pianist now, because I have played so many scales!"
If you forget about yourself when you play, including worries about how good you are and how good you will be, or how good someone else will think you are or how long it "should" take to learn this and that, if you just immense yourself in piano playing and get lost in the moment, then I'm sure that you can go as far as you will ever want. I wish you the best of luck.