In the 60's and 70's in the US, it became trendy among the upper class and celebrities to say it that way. It meant "let's get together again at some (indefinite) point in time." Today, it's often used satirically to make fun of how pretentious celebrities are. The idea is that while on the surface it sounds like an invitation, it can really be a brush-off in disguise since there isn't actually a specific invitation expressed.

It is correct, but very informal.

"Do" is used in so many expressions it would not be possible to count them all. For example, you can "do the boogie-woogie" whilst it's more accurate to "dance the boogie-woogie". On the other hand, it sounds awkward to "do baseball" or "do singing". Awkward, but not wrong! These are not mistakes; they just aren't spoken much. In general, whenever you say "do (insert any noun here)", you will not be making a mistake so long as your meaning is understood. In most cases though, there is a better word to use: "play baseball" or "sing" or "dance the boogie-woogie" or "eat lunch".

People typically say "have lunch" or "meet for lunch".

More Likely "Take Lunch" or "Take a Lunch"