Wow, you're younger than I thought you were. I guessed you to be at least 16. Oh well.
I'm no expert, but I'd say you were at a great age to start.
See if your Mom can get you to a endocrinologist, they're the best one to be able to see if you're able to be put on T. You'd have to undergo blood work but that's to be expected. Anyone undergoing HRT has to take them every few months to make sure that everything's in working order.
You're at a decent height. I've talked to many men that are shorter than you, and they're older than you are. And don't give up, you may not be done growing yet. Testosterone doesn't stunt your growth, if you're still growing, it might actually make you taller.
Again, you need to take this up with your Mom as well as your doctor. Have your Mom take you to a gender therapist, it sounds like they would do you some real good with dealing with dysphoria.
And no, you're not "too late" to have effects from T. I'm 55 and I've been on it since a month and a half before my 55th birthday. I have had definite changes even at my age. My voice has dropped to a nice baritone, I have facial hair, and my body fat and facial fat is being redistributed. The only thing that won't change is my height, but at 5'7 & 1/3"/ 171cm tall, its not too bad a height, even if I'm the shortest man in my family.
So don't think you're "too late" to go on T at your age. Good luck!
Ryuichi