Hi Anita, and welcome.
Reflexive verbs are a hard thing to wrap your mind around. Let me just give you some examples, based on the examples you gave. I hope it helps.
se me olvidó la carta / se me olvidaron las cartas.
The verb agrees with the subject, as you know, but the subject in this case is the letter/letters. This is like saying the letters were forgotten, and you are the innocent bystander. This is the normal way of saying you forgot something. If you want to emphasize the fact that you are responsible for forgetting the object, you could say:
olvidé la carta / olvidé las cartas.
Some corrections:
me olvidó el libro. 
se me olvidó el libro.
olvidé el libro.
me torció el tobillo - he/she sprained my ankle
me torcí el tobillo / se me torció el tobillo - I twisted my ankle
se me olvidó
I forgot it.
se me olvidé
me olvidó
me olvidé
(see note below about this)
me torció el tobillo
he/she/it twisted my ankle
me torcí el tobillo
I sprained my ankle.
About me olvidé,
olvidar + object is to forget something
olvidarse de + object is to forget about something. They are not used interchangeably.
Olvidé mi abrigo / se me olvidó mi abrigo - I forgot my overcoat.
Me olvidé de él - I forgot about him.
Quote:
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When do I need the se? when do the reflexive pronouns and the verb endings agree? and in general, when do I need a reflexive form. I hear it used in situations I didn´t realize it could be used.
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One thing to remember is that there are two things going on when you see the pronoun
se. It could be a reflexive verb (me torcí el hombro / me lavé las manos.) where the actor and the acted upon are the same person, or it could be a sort of passive tense like this:
Escribí una carta - I wrote a letter.
Escribió una carta - He wrote a letter.
Se escribió una carta - A letter was written.
Fue escrita una carta - A letter was written.
....
Did any of that make any sense?