All the sentences are fine, yes.
And yes, in both 3 and 4, "the lunch" means that particular meal that the two people just ate at a restaurant or cafe.
You can either keep "the" with "lunch" or omit it -- either way works fine.
1. Lunch is on me.
2. What do you mean by "lunch is on me"?
3. I mean I will pay for the lunch.
4. Thanks a lot. I really enjoyed the lunch. Then ice cream is on me.
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In #3, why is 'the lunch' used?
In #4, 'the lunch' is used. Why is that? Is 'the lunch' mean the specific lunch which was payed by the other person?
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