When do I use certain characters?

Maf, you're probably right on the former.
Even drunken Poles seem to my ears to pronounce certain sounds clearyLOL

As far as my doubts on final genitive sing. endings, there appears little rhyme or reason for "komputerA" not "komputeru" vs. "krajU" and not "kraja" etc....... Undoubtedly, it has to do with the final letter of the noun in question, "komputeR" and the like end in an "r", or something like that.

Even my first native Polish teacher never explained it!!

It's lexically conditioned, sort of like different plural endings in German. Learn the most forms and then guess when you need to.

A few nouns (most famously krawat) seem to be able to take either....

Also -u tends to block facultative animacy (treating an inanimate noun as if it were animate).

Appreciate the post, Maf.
Yeah, makes sense....sortaLOL

Yet, for example "stol" is masculine inanimate (as opposed to virile animate, e.g. "gosc"), but nonetheless takes "u" in the genitive singular "do stolU"!

Still think it must have something to do with phonemes and with what vowel the noun ends in:-)