(i) Since B is the midpoint of AC, we have:

\(\frac{2 + x}{2} = n \quad \Rightarrow \quad x = 2n - 2\)

\(\frac{m + y}{2} = -6 \quad \Rightarrow \quad y = -12 - m\)

Thus, the coordinates of C are \((2n - 2, -12 - m)\).

(ii) The line \(y = x + 1\) passes through C, so:

\(-12 - m = (2n - 2) + 1\)

\(-12 - m = 2n - 1\)

\(m + 2n = -11\)

The line is perpendicular to AB, so the slope of AB is the negative reciprocal of 1:

\(\frac{m + 6}{2 - n} = -1\)

\(m + 6 = -2 + n\)

\(m - n = -8\)

Solving the equations:

\(m + 2n = -11\)

\(m - n = -8\)

Subtract the second from the first:

\(3n = -3\)

\(n = -1\)

Substitute \(n = -1\) into \(m - n = -8\):

\(m + 1 = -8\)

\(m = -9\)

Thus, \(m = -9\) and \(n = -1\).