I find the distinction that MacMillan makes between not to mention and the supposedly synonymous let alone and still/much/even less useful: The phrases let alone and still/much/even less reinforce a negative or unlikely statement that precedes them. The still/much/even less constructs reinforce the negativity of the preceding phrase by subtraction -- Negative statement, still/much/even less ...
I think that "Let A be a set, let B be a group, and let C be a number." is the most formal phrasing. Since this is a mathematically formal usage, I think that would be preferred, but I don't think any of the phrases you presented are wrong, and "Let A be a set, B a group, and C a number." is shorter and more succinct. On a separate note, we have a sister site for Math you might consult.
As a rule, we don't use the passive voice with "let". "Allow" or "permit" is normally used instead: We were allowed to do whatever we wanted. We were permitted to drive the vehicle. Accordi...
Let go or be dragged. This is a saying often associated with Zen Buddhism (occasionally Stoicism). As far as I can tell, there's no historical connection, and it might well be originally English. ...
quotes - Origin of the saying "let go or be dragged" - English Language ...
Let’s is the English cohortative word, meaning “let us” in an exhortation of the group including the speaker to do something. Lets is the third person singular present tense form of the verb let meaning to permit or allow. In the questioner’s examples, the sentence means to say “Product (allows/permits you to) do something awesome”, so the form with lets is correct.