OPEN Total Even dominating sets:

Each vertex must have even number of neighbors,
not counting itself. This requires a NON-ZERO number
of neighbors.

These tend to have pretty patterns such as

11101111111111111111
10101000000000000001
11101011111111111101
00001010000000000101
11111010111111110101
10000010100000010101
10111110101111010101
10100000101001010101
10101111101001010101
10101000001111010101
10101011110000010101
10101010010111110101
10101010010100000101
10101011110101111101
10101000000101000001
10101111111101011111
10100000000001010000
10111111111111010111
10000000000000010101
11111111111111110111

The following grid dimensions have these:

2 by 2

3 by 3

4 by 4

5 by 2

6 by 6

7 by {3, 7}

8 by {2, 8}

9 by {4}

10 by 10

11 by {2, 3, 11}

12 by 12

13 by {6}

14 by {2, 4, 14}

15 by {3, 7, 15}

16 by 16

17 by {2, 8}

18 by 18

19 by {3, 4, 19]

20 by {2, 6, 20}

Total Even dominating sets:

Each vertex must have even number of neighbors,
counting itself. This requires a NON-ZERO number
of neighbors.

The following grid dimensions have these:
2 by {1, 3}

3 by {2, 5}

4 by 4

5 by {1, 2, 3, 5, 7, 11}

6 by 8

7 by {2, 5, 11}

8 by {1, 6, 13}

9 by {2, 4, 14}

11 by {1, 2, 3, 5, 7, 11, 15}

13 by {2, 8}

14 by {1, 4, 9, 16}

15 by {2, 11, 23}

16 by 14

17 by {1, 2, 13}

18 none

19 by {2, 4}