I've been doing various forms of martial arts on and off at various ages in my life. Tae Kwon Do and Karate when I was 12 for about 1 and 1/2 years. I attempted judo when I was about 14. I did kendo for some time when I was 17. And now, at age 21, I am looking to do Northern Shaolin kung-fu.
To answer your question, you're as young as you feel! Don't let age be a factor when it comes to pursuing something new!
As for the particular types of martial arts, I can say that anything physical can help with anxiety and depression. I always felt much better after doing anything pertaining to martial arts. Most of them are very heavy workouts, so be prepared to be sweating, hot, and exhausted after your first go! Tae Kwon Do and Karate vary per instructor. I once had a Tae Kwon Do instructor who made us run around in circles for 10 minutes each class. That wore us out and made us dizzy moreso than it taught us the martial art. Another note on that, be careful of which instructor you choose! Some are just simply in it for the money!
One more note is make sure you find a type of martial art that's meant for you! Many martial arts are very physical and some are very aggressive. Each type has a specific teaching goal. Here's my experience:
Kendo is a Japanese swordfighting type of martial art. It involves sparring (yes, you can whack your opponent with a bamboo sword when you get to the level) and is quite fun!
Judo is extremely aggressive, as it involves you attempting to bring down an opponent larger than you by targeting their weak points. Last I remember, I ended up on the floor a few times.
Northern Shaolin is a form of kung fu. It involves a lot of leg movement and is a rather aggressive form of martial art. This is my most recent one that I hope to join after I get enough money to do so!
If you want something more relaxed, I suggest trying Tai Chi. It's supposedly a more flowing, relaxing form of martial art. Or yoga, though that's not technically a martial art
My apologies for the large amount of text! I love martial arts (if you couldn't tell, lol). I hope you find what works best for you when it comes to a sort of physical activity!