All this does is linearly adjust the normally-flat 5% probability for each number to occur. What results is a increased or decreased probability of any number above or below average to occur, positively for advantage and negatively for disadvantage. See this AnyDice function set, which yields the following:

^{Black is d20, orange is highest of 2d20, blue is lowest of 2d20.}

Since the probability of achieving any given number is a linear function, we can use linear regression (via Wolfram Alpha and our sample data from AnyDice to eventually solve for probability of x = 0.5x - 0.25 - multiply by 100, and there's your percent chance that you'll roll any particular number.

Additionally, what you're likely looking for is the probability that at least a particular number will be rolled, using either advantage or disadvantage. AnyDice, again, is king:

^{Black is d20, orange is highest of 2d20, blue is lowest of 2d20.}

Data:

Advantage
#     %
1     100
2     99.75
3     99
4     97.75
5     96
6     93.75
7     91
8     87.75
9     84
10    79.75
11    75
12    69.75
13    64
14    57.75
15    51
16    43.75
17    36
18    27.75
19    19
20    9.75

Disadvantage
#     %
1     100
2     90.25
3     81
4     72.25
5     64
6     56.25
7     49
8     42.25
9     36
10    30.25
11    25
12    20.25
13    16
14    12.25
15    9
16    6.25
17    4
18    2.25
19    1
20    0.25

No, they don't auto-succeed

Crit successes and failures only apply to attack rolls (PHB p. 194, Attack Rolls subsection).

Even though grapples and shoves are a "special melee attack" (PHB p. 195), they are not resolved by making attack rolls. Therefore, the specific rule regarding crits does not apply to them.