For 1) and 2) I agree with Isi's response. "Monte" in spain can also refer to a general area with hills or mountains (for example "mi padre se ha ido al monte a cazar" - that doesn't mean he went to a mountain, but just to some area with nature, it could just be a forest or some hilly area)
For 3) they're all similar, all cascara, piel and corteza are protective membranes. In my mind:
- "cascara" is a hard surface that can crack (like a egg, or a walnut... in fact if you give 5 seconds to think about "cascara", everyone will say egg or walnut).
- corteza is a a hard/rough/uneven surface that doesn't crack open like a cascara. Corteza de un arbol, is the most typical example. Also, corteza terrestre and corteza cerebral (Brain). Note how all those are not smooth, and don't crack like an egg or a walnut.
- piel is for something that is not hard, like cascara or corteza. Usually smooth or furry. Piel de animales, de platano, de naranja, etc.
There are cases which are borderline... for example I would say that a pineapple has corteza, and a watermelon has piel, but if someone used piel/corteza instead, it wouldn't necessarily strike me as too odd.
Said all that... I don't think that with many of these synonyms you can derive a clear RULE to follow, because generally there isn't one, you just have to learn by looking at the examples and contexts where those words are used, and accept that there is a certain amount of overlap between synonyms.