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.