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LingakuLingaku～ませんか invites someone to do something: "won't you…?", "would you like to…?", "shall we…?". Take a verb in its ます-form and change ます to ませんか.
It looks like a question about not doing something, but as an invitation it isn't one. Nobody is asking whether you don't drink coffee. The negative makes the invitation polite, because it leaves the other person room to say no.
～ますか asks for a fact. ～ませんか invites. ～ましょう is "let's", and takes the yes for granted.
English invites with "do you want to…?". In polite Japanese that does not work: ×一緒に映画を見たいですか asks about the listener's wishes and proposes nothing. Use ～ませんか.
To accept:
A no is seldom said outright. Start with すみません and give a reason, or trail off after ちょっと and let the listener fill in the rest.
いいえ、いきません行きません is correct Japanese and a flat refusal.These go beyond the basic invitation.
Between friends the same invitation is the plain negative said with a rising tone.