Keeping this in view, will you go out for dinner?
To go out to dinner usually means that one is invited somewhere, so it's not something one can spontaneously undertake. To go out for dinner usually means that one decides to go out, rather than stay in and prepare food at home, For these reasons I'd be more likely to say 'Let's go out for dinner'.
Beside above, what to say to someone who is going to eat? Is someone is eating any meal at all and you wish to say something positive, you can always say, “I hope you enjoy your meal!” — if you are leaving. Or “I hope you are enjoying your meal.” — if you are stopping to chat with someone and will linger for a moment.
Also, is it out to lunch or out for lunch?
"Out to lunch" simply means going out (of your place of residence) for the purpose of having lunch. Here, the word lunch is used in the verb form as opposed to its noun form which indeed is a shortened version of luncheon.
What does it mean if someone is out to lunch?
The expression out to lunch is an idiom that means "too strange or confused to know what is really happening." Although some idioms have parts that can vary, this one is fixed. The idiom out to lunch is almost always used with the verb to be, as in these examples: I thought to myself, "She's completely out to lunch."
