gemma pell gemma pell, et toi?

Why couldn't you use it?
I am Gemma Pell. You can say that in other languages, why not French?

because it is incorrect in france

Nope, it's correct.
When someone ask "Qui est Yukihaki?" (Who's Yukihaki?) You can say "Je suis Yukihaki" (I am Yukihaki)
It's the "simple" way to introduce yourself

Has that teacher ever been in France? I often say "Je suis Marie-Rose" to introduce myself.

Teachers tell you not to, because it's not the most elegant way to express yourself

That's correct. The m is there because the phrase would technically be je me appelle, which directly translates to "I call myself." However, the French like their language to be fluid and pretty, so they drop that extra vowel syllable and just run together the two words, resulting in je m'appelle.

*five years of french paid off, yes*

Correct! From a french girl here :)