Unique Pairs and a Domino Square

1. How many different digits are needed to fill an NxN grid, such that every pair of adjacent digits (horizontally and vertically) is a unique pair? Solve for N=2,3,4,5,6. Give examples. For example, for N=2, three digits are needed:

   00 12

   This 2x2 grid has the four unique pairs 00, 01, 02, and 12.

2. How many different digits are needed to fill an NxN grid, such that no pair of digits can be found adjacent in the grid more than twice? Solve for N=2,3,4,5,6.
3. How many different digits are needed to fill an NxN grid, such that no pair of digits can be found adjacent in the grid more than three times? Solve for N=2,3,4,5,6. My extended question for this is, Can a set of double-5 dominoes (minus 3 dominoes) fill a 6x6 grid such that no pair of digits can be found adjacent more than three times?

Source: Original.

An NxN board will require N*(N-1)*2 adjacencies
2x2:2*1*2=4  adjacencies
3x3:3*2*2=12 adjacencies
4x4:4*3*2=24 adjacencies
5*5:5*4*2=40 adjacencies
6*6:6*5*2=60 adjacencies

D digits will produce Sum(1 to D) combinations
0- 0:1=1 combinations {00}
0- 1:1+2=3 combinations {00,01,11}
0- 2:1+2+3=6 combinations {00,01,02,11,12,22}
0- 3:1+2+3+4=10 combinations {etc.}
0- 4:1+2+3+4+5=15 combinations
0- 5:1+2+3+4+5+6=21 combinations
0- 6:1+2+3+4+5+6+7=28 combinations
0- 7:1+2+3+4+5+6+7+8=36 combinations
0- 8:1+2+3+4+5+6+7+8+9=45 combinations
0- 9:1+2+3+4+5+6+7+8+9+10=55 combinations
0-10:1+2+3+4+5+6+7+8+9+10+11=66 combinations

1. Limited to one occurrence of each adjacency.
  N=2, D=3, A=4:{00,01,02,12}
  00
  12

  N=3, D=5, A=12:{01,02,03,04,11,12,13,22,24,33,34,44}
  113
  203
  244

  N=4, D=7,
A=24:{00,01,02,03,04,05,06,12,13,14,15,16,23,24,25,26,34,35,36,44,45,46,55,5
6}
  4455
  3006
  5123
  2461

  N=5, D=10,
A=40:{00,01,02,03,04,05,06,11,12,13,14,15,16,17,18,19,22,23,24,29,

33,34,35,36,39,45,46,48,49,55,56,57,58,66,67,68,69,78,89,99}
  42339
  92046
  91055
  81671
  53684

  N=6, D=12, A=60:{00,01,02,03,04,05,06,07,08,11,12,13,14,15,16,17,1a,
                   22,23,24,25,26,27,28,2a,33,34,35,36,37,38,39,
                   44,45,46,47,4a,4b,55,56,57,58,5a,5b,66,67,69,6a,
                   77,78,79,7a,7b,88,89,8a,99,aa,ab,bb
                   (where a=10 and b=11)

  17bbaa
  a55447
  660033
  411228
  253678
  a89970

2. Limited to two occurrences of each adjacency.
  N=2, D=2, A=4/2:{00,01,01,11}
  00
  11

  N=3, D=3, A=12/2:{can't be done}
  N=3, D=4, A=12/2:{00,00,01,01,02,11,12,12,13,23,23,33}
  000
  112
  233

  N=4, D=5, A=24/2:{00,00,01,01,02,02,03,03,04,04,11,12,
                    13,14,14,23,24,24,33,33,34,34,44,44}
  3112
  4003
  4203
  1443

  N=5, D=6,  A=40/2:{00,00,01,01,02,02,03,03,04,04,05,05,
                     11,11,12,12,13,14,14,15,15,22,22,23,
                     23,24,25,25,33,33,34,34,35,35,44,44,
                     45,45,55,55}
  44553
  10003
  11432
  22435
  02515

  N=6, D=8, A=60/2:{00,00,01,01,02,02,03,03,04,04,05,05,06,06,07,
                    11,11,12,13,13,14,14,15,15,16,17,22,22,23,25,
                    26,26,27,33,33,34,34,35,35,36,36,37,44,44,45,
                    45,46,46,47,47,55,55,56,56,66,66,67,67,77,77}
  655566
  111226
  400027
  453337
  431460
  067745

3. Limited to three occurrences of each adjacency.
  N=2, D=2, A=4/3:{00,01,01,11}
  00
  11

  N=3, D=3, A=12/3:{00,00,01,01,01,11,11,12,12,12,22,22}
  000
  111
  222

  N=4, D=4, A=24/3:{00,00,00,01,01,01,02,02,03,11,11,12,
                    12,12,13,13,13,22,23,23,23,33,33,33}
  0000
  1112
  2233
  0331

  N=5, D=5, A=40/3:{00,00,00,01,01,01,02,02,02,03,03,03,04,04,
                    11,11,11,12,12,12,13,13,13,14,14,22,22,23,
                    23,23,24,24,24,33,33,33,34,44,44,44
  00001
  22211
  33312
  01444
  33042

  N=6, D=6, A=60/3:{did not find any yet}

 _ _ ___ _ _
|0|0|0 0|1|0|
|1|2|2 2|1|3|
|1|1 3|2 3|0|
|4|4 3|4|3|5|
|1|5 5|0|5|5|
|5|2 4|4 4|2|