(i) Between and among. The Oxford English Dictionary tells us not to take too seriously those who tell us that between must only be used of two things and that when there are more the preposition must be among. It says:
Between is still the only word available to express the relation of a thing to many surrounding things severally and individually, among expressing a relationship to them collectively and vaguely: we should not say "the space lying among the three points", or "a treaty among three powers" or "the choice lies among the three candidates in the select list", or "to insert a needle among the closed petals of a flower".
Between each. Grammarians generally condemn the common use of between with each or every, as in "there will be a week's interval between each sitting". It is arguable that this can be justified as a convenient way of saying " between each sitting and the next", and that, considering how common it is, only pedantry can object. But those who want to be on the safe side will say either "weekly intervals between the sittings" or "a week's interval after each sitting".
Between . . or and between . . . and between. If between is followed by a conjunction, this must always be a simple and. It is wrong to say: "the choice lies between Smith or Jones", or to say "we had to choose between taking these offices and making the best of them and between perhaps finding ourselves with no . offices at all". If a sentence has become so involved that and is not felt to be enough it should be recast.