### Part (a)

We need to prove that for $n = 2001$ and $k = 4001$, it is not possible to color the cells of a $2001 \times 2001$ chessboard with 4001 different colors such that for every $i \in \{1, 2, \ldots, 2001\}$, the $2n-1$ cells on the $i$-th row and $i$-th column have all different colors.

1. Total Number of Cells:
The total number of cells in a $2001 \times 2001$ chessboard is:
$$(2001)^2 = 4004001$$

2. Cells on the Main Diagonal:
The number of cells on the main diagonal is:
$$2001$$

3. Cells Not on the Main Diagonal:
The number of cells not on the main diagonal is:
$$4004001 - 2001 = 4002000$$

4. Pigeonhole Principle:
Since we have 4001 colors and 4002000 cells not on the main diagonal, by the pigeonhole principle, there must be at least one color that appears in at least:
$$\left\lceil \frac{4002000}{4001} \right\rceil = 1000$$
cells.

5. Contradiction:
If we have a set $M$ of at least 1000 cells colored with the same color, then there must be at least two cells in $M$ that are either in the same row or the same column. This contradicts the requirement that all cells in the same row and column must have different colors.

Therefore, it is not possible to color the $2001 \times 2001$ chessboard with 4001 colors under the given conditions.

$\blacksquare$

### Part (b)

We need to show that for $n = 2^m - 1$ and $k = 2^{m+1} - 1$, such coloring is possible.

1. Constructing the Chessboard:
Let $n = 2^m - 1$. We need to color the $(2^m - 1) \times (2^m - 1)$ chessboard with $2^{m+1} - 1$ colors.

2. Latin Square Construction:
A Latin square of order $n$ is an $n \times n$ array filled with $n$ different symbols, each occurring exactly once in each row and exactly once in each column. For $n = 2^m - 1$, we can construct a Latin square using $2^m - 1$ symbols.

3. Orthogonal Latin Squares:
We can construct $2^m - 1$ mutually orthogonal Latin squares (MOLS) of order $2^m - 1$. Each of these Latin squares can be filled with $2^m - 1$ different symbols.

4. Combining Latin Squares:
By combining the $2^m - 1$ MOLS, we can create a $(2^m - 1) \times (2^m - 1)$ array where each cell contains a unique combination of symbols from the $2^m - 1$ MOLS. This gives us $(2^m - 1)^2$ unique combinations.

5. Color Assignment:
Assign each unique combination of symbols a unique color. Since we have $(2^m - 1)^2$ unique combinations and $2^{m+1} - 1$ colors, we can assign each combination a unique color.

6. Verification:
Each row and each column will have all different colors because each row and column in the MOLS are filled with different symbols.

Therefore, it is possible to color the $(2^m - 1) \times (2^m - 1)$ chessboard with $2^{m+1} - 1$ colors under the given conditions.

$\blacksquare$