Apologies for yet another answer, but I really think I finally got it, as it fits all clues and your bonus hints!

> My first semester—no, my entire freshman year—in high school, I was all alone.

>! In semester $n$ you are the identity in $A_n$. $A_1$ and $A_2$ don't have any other elements.

> As I started my sophomore year, I finally made a friend. Two, actually. 

>! Your new friends are the other two elements of $A_3$

>They weren't in the same class as me, but for the rest of my high school life, they and their classmates basically defined my whole friend group.

>! They are not in your conjugacy class, but they and their classmates, together all 3-cycles (there are one or two conjugacy classes of 3-cycles, these two represent both classes), generate your whole friend group, namely $A_n$, which is generated by its 3-cycles.

> As sophomore year drew to a close, I made yet more friends, from another class. There were finally some dynamics between us.

>! Next semester, more friends: $A_4$. Some from another class, namely that of the products of disjoint transpositions. Not sure about the dynamics.

> Cue the start of junior year. More friends from yet another class, yet I feel things won't be so complicated anymore. (This semester we were big into D&D, but we had to substitute d12's for d20's).

>! Not so complicated anymore, namely simple: $A_n$ for $n \ge 5$ is simple. The new class(es) are the two classes of 5-cycles. $A_5$ is the symmetry group both of d12 and of d20.

> At some point during my sixth semester, we looked at ourselves in the mirror and realized we were quite exceptional.

>! In the mirror you saw all your symmetries, those of $A_6$, which are exceptional: all other alternating groups have an outer automorphism group of (1 or) 2 elements, but that of $A_6$ is the Klein 4-group.

> Start of senior year: Exponent? Blaze it!

>! The exponent of $A_7$ is 420. This is slang for smoking marihuana.

> How many friends did I have by graduation, and across how many different classes?

>! You have $8!/2 - 1 = 20159$ friends, across (if I counted correctly) 13 conjugacy classes, including your own.