My friend Alex from school works at a grocery store. Its also the same grocery store that my cousin works at, and I'm not out to my cousin yet (even tho I really should be by now). Alex figured out that we were related after seeing a photo I had on facebook. He decided that he would talk about me to my cousin and mention that we go to the same school Alex used my preferred name when talking about me, as well as male pronouns because he doesn't know otherwise. I'm guessing my cousin got confused and told him what my real name was, although I'm not sure if my cousin mentioned gender 

I came to school today unaware of all of this, until in my first class I could hear my other friend calling out my birth name from across the room, trying to get my attention  I ignored him but he called out to me, "do you know anyone called (my birthname)?" We were in the middle of class so I lied and said that I had a cousin called (birthname). He turned to his friend and said "well Alex is an idiot, he doesn't know what he is talking about". After this, I had found out that Alex told a couple of people about how my cousin told him my real name.

At the moment everyone thinks Alex is pretty stupid for "getting mixed up" and actually saying that he thought I had a girls name. But he knows what my cousin has told him is probably true and I have a feeling he will try to prove the truth to everyone just so he doesn't look bad anymore. I'm actually pretty scared for what might come. Its driving me crazy, I don't know how to answer these harder questions and Ive used so many excuses before. I'm thinking about just coming out, but I'm not sure how well that will go. Ive found that I don't act myself around people who know that I am trans, I'm not sure why but it is probably a bad thing. I dont want anyone to make me physically prove I am a boy either, that would be a nightmare Alot of my friends are bigger and stronger than me so there would be nothing I could do!

Okay, sounds like you need to take a deep breath.  lol.  Not that it will fix anything, But I can seriously feel your stress through your words. 
Don't think about worse-case scenarios.  that never helps, lol.

I see you have a few options.  I don't know all the people involved, so I don't know which is better.
1) I think either you come out to your cousin and have her/him pretend to Alex that he/she lied about your name to trick.  Or if your cousin's not cool enough to do that for you,
2) Come out to Alex and tell him why you don't want him going around and spreading their name thing out.  but Idk, maybe Alex isn't cool enough for that either.

Here I think is your options for keeping it all hidden.
3)  Tell Alex that your parents gave you a girl's name because they hate you.  and it's really embarrassing to you and you're changing it when you turn 18, and you'd like for him to not go embarrassing you to everyone, please.
Or,
4)  Pause for a minute and really think about what you would do if you were born cis-gendered male, and your cousin (or Alex) lied about your birth name being a girl's name.  You would totally laugh, wouldn't you?  You might even go along with it, you might be all like, "Yeah, my parents hate me, so they gave me a girl's name."  And if you make it sound like you're lying, your friends could tell you were lying, and so they would be like, "You're lying, you don't really have a girl's name.  I don't believe you."
In other words, if you defend yourself, they're think you got something to hide.  but if you jump right in and "jokingly" say that that is your name, they'll think since you're joking about it that it can't be true.

All completely amazing advice from JohnAlex, I agree with everything he said. Most importantly though is try and chill a little, stressing won't help anything and could just make you look more 'guilty' to those involved.

Good luck mate!

I would just play it off like your cousin was just being a dick and trying to mess with your life, to fix everything with the general public, then go individually to Alex and come out to him. I think in this situation, that you should keep your radar up and make sure you don't answer to your birthname, because that would really give you away and would have people scratching their heads.