First, let's correct the small spelling error in your otherwise lovely sentence:
Me pidió que le llamara.
Note that the verb pedir is conjugated like this for the third person singular past tense: pidió.
Now for your questions.
1) Why is a subjunctive used here?
There is something similar in English: She asked that I be more assertive. Note that we don't say: She asked that I am more assertive.
There may be technical explanations possible, but the most practical way of looking at this is that the linking word "que" leads to a subjunctive, in Spanish, most of the time.
2) Why does "pedir" denote a request and, for instance, "decir" doesn't? What determines whether or not a verb is used as a request?
First, "pedir" denotes a request because "pedir" means "to request." (See Collins.) (Notice that "ask" has two basic different meanings in English: one is to request, and the other is to ask a question.)
Now, as to whether "decir" can be used as part of a request, I would say that in informal speech, yes, it could, I suppose; but if you want to make sure you're being polite, "pedir" would be a better choice. However, context matters. "His mother asked him to straighten his room before leaving" is technically a request, but I have a feeling she doesn't consider it optional!
I would think that common sense and context will tell you whether something is a request; but you can always ask ("¿Me dices o me pides que lo haga?"). But I have a feeling there's something you want to understand about this that I haven't quite understood.
3) How would the meaning change if it was "llama"? Or "llamaría"? Or "llamará"?
First, I'd like to make sure it's clear to you that in this sentence, everything is in the past tense. "Pidió" is simple past, and "llamara" is past subjunctive. Let's see what this would look like if we shift it to the present:
Me pide que le llame. | He is asking me to call him / He is asking that I call him / He is requesting that I call him.
None of the other three verb forms could be substituted into this particular sentence in place of "llamara." Here are some sentences that would work:
Si le pido, me llama. | If I ask him to, he will call me.
Yo de tí, le llamaría. | If I were in your shoes, I would call him.
Mi tía le llamará mañana. | My aunt will call him tomorrow.
Each verb form has its own uses.