For question 2, I have an answer that I think is optimal
Toss dice 1811 and 1907. (1811*1907)%2022=1 so you can divide the possibilities into equal piles with only 1⁄3453577 chance of getting the leftover. If you do get the leftover, repeat the procedure. Expected number of roles 2.000000579
Note that you must always toss at least 2 dice to make this game fair. Any strand of possibility where you only have one dice is too likely.
I bruteforce searched for pairs of primes to minimize (p*q%2022)/(p*q) (the chance of needing to repeat) and these were it.
For question 2, I have an answer that I think is optimal
Toss dice 1811 and 1907. (1811*1907)%2022=1 so you can divide the possibilities into equal piles with only 1⁄3453577 chance of getting the leftover. If you do get the leftover, repeat the procedure. Expected number of roles 2.000000579
Note that you must always toss at least 2 dice to make this game fair. Any strand of possibility where you only have one dice is too likely.
I bruteforce searched for pairs of primes to minimize (p*q%2022)/(p*q) (the chance of needing to repeat) and these were it.