For part (a), observe that if a square filled with the number $x$ has an adjacent square with a number greater than $x$, then it must also have an adjacent square with a smaller number. Hence, if there exist 2 adjacent squares with different numbers, there must be an infinite sequence of adjacent squares so that the numbers in the squares are $a_1>a_2>a_3>\dots$. However, since $a_i$ are nonnegative integers and have a lower bound, this is impossible. For part (b), choose an arbitrary row in the plane, and write all integers in the squares of that row increasing from left to right. Then, for every column let all the squares in the column have the same number. This configuration has both $2024$ and $2024^{2024}$.