I just finished eating. - should be Present Perfect here (but in AmE Past Simple is also possible)
I already finished eating. - same as for the 1st sentence
Once you finish eating / or Once you've finished, come over here.
After I've finished eating, I'll come over. - You can also say, "When I finish.."
Can't you just wait until I finish eating? - both sentences (including the next one) are correct
Can't you just wait until I've finished eating? -
Have you finished eating your lunch / dinner?