Brad's examples are perfectly correct, but I'm sure you've noticed that American speech rarely if ever resembles correct written grammar. If your goal is natural-sounding conversation, Ben's examples are fine -- "Have you had lunch?" is a correct and natural way to say it. In America, you're also likely to hear something like, "Did you have lunch yet?" Though probably grammatically questionable, it's commonly said and nobody would find it unusual. For the second, "Sorry, I've already eaten," is also correct. Again, there's a lot of variation (at least in my experience) with American speakers. To sound natural when you're talking with your colleagues, you can try out phrases that you hear them use. I wouldn't advise this for someone who isn't immersed in English the way that you are, but in your case I think it's okay to choose to speak using "bad" grammar if it helps you to fit in more comfortably with the people around you. As long as you can remember what the correct grammar is, there shouldn't be a problem -- the same way that chatting with your friends in casual Japanese doesn't make you forget to speak respectfully to your supervisor at work. 頑張ってね　＾＾

In London, it's nearly always "Have you had lunch?" "Eaten" is rarely used. "Let's have lunch together." "I've had lunch already."

The best answer is, "Have you eaten lunch yet?" The present perfect form is used to show a period of time from the past until right now, and that's what the speaker really wants to know. Asking "Did you eat lunch" means at some time in the past (yesterday? last week?), but it doesn't include the time that continues until right now. We don't use "do" with actions like eating: we use the various verb forms of "eat." Do forms are usually used for tasks, so it sounds very unnatural to use it with actions like eating, sleeping, speaking, etc.

They are all fine. I would say whatever you think of first is natural.