From what I can gather, this has the same scentence structure French does, without all of the words. (The house of my friend vs. My friend's house) They just say "She student my friend". I guess of you look at it and see that this is what the sentence portrays you can work it out?
Possession in Russian is expressed through the genitive case -- моего друга is genitive, so this sentence is absolutely unambiguous. You can look at it in opposition to, for example, "He's my [female] student's friend" --> Он друг моей ученицы. You're not entirely wrong in your comparison to French, as the genitive case here fills the same role as the preposition "de" does in "C'est l'ami de mon élève." -- you simply were missing a crucial bit of information :-)
Of course kpagcha's proposed translation is also correct, and it's almost certainly been added to the possible answers by the mods since then.
And could you use "своей" and "своего" instead of "моей" and "моего"? Why or why not? I have absolutely no clue of when to use these.
Свой and its declensions always refer to the subject of the sentence as the possesser (the subject is the noun or the pronoun in the nominative case), so here «Она ученица своего друга.» would mean "She is her [own] friend's teacher."
It's also true when you or I are the subject of the sentence, and this is where it gets hairy for us English speakers. For example, "I am my brother's keeper." in Russian becomes «Я хранитель своего брата.» and not «Я хранитель моего брата.» Likewise, "Mind your own business." becomes «Занимайся своим делом.» and not «Занимайся твоим делом.»
Sie ist die Schülerin meines Freundes? Damn it doesn't help that German Genitiv feels weird
's Uncle twice removed's brother's girlfriend's sister's dad's boss's aunt's gramma...
I find this sentence very strange for some reason... Is this the best way in Russian to construct that meaning? "She is my friend's student"
Студент/студентка refers specifically to a student of an university, academy, college etc. Someone who attends any sort of higher educational establishment.
Ученик/ученица - a pupil, a learner or a student in general sense.