I've hit an impasse with problem 1 (squares whose only decimal digits are 1 and 0).

Apparently unsolved. Best I could do was

OEIS A016070, Numbers n such that n^2 contains exactly 2 different digits, excluding 10^m, 2*10^m, 3*10^m.

Gives 21 terms, "No other terms below 3.16*10^20" Sequence thought to be finite.