Invitations, suggestions, and offers
ませんか, ましょう, and ましょうか invite different kinds of responses. Consider who will act and how much agreement the speaker is assuming.
Invite with ませんか
一緒に昼ご飯を食べませんか。 means 'Would you like to have lunch together?' Although it looks like a negative question, it invites participation rather than merely checking that someone will not eat.
A specific time or activity makes an invitation easier to answer. 土曜日に映画を見ませんか。 invites someone to see a movie on Saturday. Whether ませんか is an invitation still depends on the verb and situation.
Suggest with ましょう
Replace ます in 食べます with ましょう to make 食べましょう. じゃあ、ここで食べましょう。 means 'Then let's eat here,' suggesting a joint action or moving ahead with an agreed plan.
If the other person has not agreed to participate, a firm ましょう can sound as though agreement is being assumed. Distinguish an initial invitation from the next step in something already agreed.
Suggest or offer help with ましょうか
荷物を持ちましょうか。 means 'Shall I carry your bags?' Here the speaker offers to carry them. It differs from 持ってください, which asks the other person to do the carrying.
そろそろ帰りましょうか。 can suggest a shared action: 'Shall we head home soon?' ましょうか does not always mean 'Shall I do it for you?'; identify the participants from context.
Check: Who is being instructed to carry the bags in 荷物を持ちましょうか。?
Show the answer and why
Normally, nobody is being instructed: the speaker is offering to carry the bags. A request for the listener to carry them would use a different expression, such as 荷物を持ってください.