Invitation (Let's / Shall We)

Suggesting something to someone who may well not want to.

Verb ませんか。 vs Verb ましょう: Use ませんか. Phrasing the invitation as a negative question leaves them room to decline, which is why it is the polite default. ましょう assumes agreement and works best once the plan is already settled, or with people you are close to.

Offering something rather than proposing to do something together.

～いかがですか。 vs Verb ませんか。: Use いかがですか. It asks after their preference about a thing (コーヒーはいかがですか), so it fits offers and hospitality. ませんか proposes a joint action and needs a verb, so it cannot offer an object at all.