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I'm kind of confused by what our teacher taught us today. He specifically said that "could" is the past form of "can", and the same for "shall" and "will". I also recall that he gave as this example:
"That kid could produce a few words when he was 2." – to mean "That kid was able to produce a few words when he was 2."
I only know that could is more polite (when it comes to permission) but is that a valid sentence?
Your teacher is quite correct: could, might, should, would are the 'past' forms of can, may, shall, will. The example your teacher gave you is perfectly OK; you would also use these forms to backshift present-tense forms in reported speech:
"I can bench-press three hundred pounds", says John.
John said that he could bench-press three hundred pounds.
Must has no distinct 'past' form; historically, however, it is the 'past' form of a verb which has lost its present form, mote.
And that's a process that's still going on today. Shall has almost disappeared from Present-Day English, except in legal contexts, and may is in steep decline. Must is very little used now as a past-tense form; some teachers even tell their students that had to must be used instead. Similarly, past-tense could and would are giving way to was/were able to and was/were going to.
I suspect that what underlies these usage shifts is that the so-called 'past' forms of verbs don't always signify backshift—past-tense reference—but are also used to signify what we might call 'sideshift': a less assertive social or logical modality. This is especially true with the core modal verbs, which are the primary indicators of modality; the 'past' forms of these verbs are used so frequently to signal social or logical distance that the use for actual past-tense reference has become secondary. Except in the most formal registers, could, might, should, would have become almost (but not quite) disconnected from their present-tense forms.
