This is very interesting. Some suggestions:
My name is [Fill in my name], but you can call me whatever you'd like until I get your name down.
What do you mean here? What does your not getting his name down have to do with his calling you whatever he wants?
I have to wake up early, because in Phoenix, waking at seven could mean running in near-hundred-degree weather.
"I used to wake up early because, in Phoenix, seven in the morning meant running in near-hundred-degree weather."
I feel that the first sentence in the second paragraph belongs in the first paragraph, where you talk about names. You can start off the second paragraph with "I love running..."
Other than that, I like calculus or physicist jokes.
Like which ones? It would make your essay stronger if you included a small example.
Other than that, I like calculus or physicist jokes. I drink lots of water for my running. I incessantly have to go to the bathroom (surprise there). I've developed immunity to caffeine from many late nights studying. I write more than I talk, although the balance is starting to swing in the other direction. I draw lots of aliens, though they're supposed to be humans. I get along with almost anyone, and I really look forward to meeting you.
You go through a lot of things here. Others may disagree, but in my opinion, I think you should focus on one thing in this paragraph. Maybe you can talk about your physics jokes, or maybe you can elaborate on your "alien" drawings of humans :) To me, focusing on one thing at a time makes an essay more fluid and easier to read.
If you don't mind, could you take a look at mine?