1.
For four rounds, here is one solution.

8 6 2 2
1 1 3 3
7 6 7 5
8 4 4 5

First round: 1, 1; 2, 2; 3, 3; 4, 4; 5, 5.
Second round: 6, 6.
Third round: 7, 7.
Fourth round: 8, 8.

A solution for five rounds:

8 5 7 8
1 1 2 2
6 5 6 3
4 4 7 3

First round: 1, 1; 2, 2; 3, 3; 4, 4.
Second round: 5, 5.
Third round: 6, 6.
Fourth round: 7, 7.
Fifth round: 8, 8.

As far as I can tell, five is the maximum number of rounds for a four by four grid, assuming the final number is finite. Intuitively, one couple can depend on a previous one only if they are separated by one or more squares that are not in a corner. That fact, however, is just the beginning of the proof. I have not found a pretty proof.

The 6 x 6 grid problem is also open.