"Can I look at my textbook?"
All of the elements have meaning. "Mo" conveys a sense of additionality (too, also), ranging to the unexpected (even, although). "Mite mo ii?" then, is something like "Looking, too, is OK?" or "Even looking is OK?" "Mite ii?" leaves this out or leaves it to be understood (which is consistent with the Japanese propensity for not saying what can be understood except to be polite).
Mite ii, mite mo ii, and mite wa ii are all possible and the difference between them is the difference in the sense or meaning of the particles. First, particles indicate but do not create grammatical function. So, "mite ii," with or without a particle, means roughly "looking is OK." Add "mo" and you get "looking, too ..." or "even looking is OK." Add "wa" and you get something like "looking, as compared or contrasted with whatever else might be mentioned, is OK." In other words the particles add nuance but the grammar of "mite ii" is constant.
います is a verb that literally means something like "to exist" and is only used for animate beings. So like, 女の人がいます means there is a woman.
いい is an adjective and です is.. something akin to "is". It essentially allows the adjective to function as the core meaning of the sentence without needing a verb. 女の人がいいです means the woman is good.