what is chaz bono
Words don't get contracted deliberately. They just become that way over time. People may start out saying "let us", but sooner or later, certain syllables will be eroded and letters will be dropped.<br />
Say any phrase slowly and carefully - all the syllables will come out. Say it quickly, or slur it and some syllables will naturally be dropped. Those are the syllables that tend to become dropped over time, as a more efficient or quicker way of speaking gradually slides into place.<br />
Apart from that, there's no real pattern to it, so no rules to follow.
Because we have traditionally lazy tongues, and contractions are easier to say.
Contractions are formed by joining two words together, omitting one or more letters: the apostrophe just stands in place of the missing letter(s), whatever they are. The one that seems to cause most confusion and incorrect use is "it's" - a contraction of "it is". Where "its" means "belonging to it", it doesn't have an apostrophe because there are no missing letters: it's not a contraction of anything.
I can only guess that it's to do with the frequency with which a particular common combination of words is used: the verb "to have" is frequently contracted - "I've" for "I have" and "he's" for "he has" (as well as, incidentally, "he is") and so on. "Not" gives rise to a whole host of contractions - "can't" "mustn't" "shan't" "wouldn't" "couldn't" ... it must all be very confusing for people trying to learn English!
It just is. I wish we had more common contractions like "a lot", a contraction for "his and/or her" for literature, "you'd" (I'm not sure if this is a legit contraction or not), "who'd" and a few others.
Let us....with may.we, or shall we?
"May we" is asking for permission. Like when children are asking for sweets or something like that.
"Shall we" is a suggestion. Like when people are trying to decide what to do with the day and someone suggested going for a walk.
Over time, however, the subtle distinctions have been discarded and people aren't always aware of them. That's why most people inaccurately start a phrase with "can we" - and in its correct form, that means asking whether a person is capable of doing something.