Trouble understanding theory for error detection and correction in repetition code
Question 1: Suppose $k$ is fixed. Then, if any group of repeated $k$ bits is flipped, the error is not detected, so we need $k \ge m + 1$. On the other hand, if $k = m + 1$, then any combination of $m$ errors will be detected because at least $k$ bits must be flipped (i.e, the whole group of $k$ repetitions) in order to arrive at a valid code word.
Question 2: The reasoning is similar. One bit errors can be fixed because of the parity bit. Two bit errors cannot because flipping the parity bit and any other repetition bit yields a word $w'$ with Hamming distance $2$ from the original code word $w$; from it, flipping the other two repetition bits yields a valid code word $w''$ which also has Hamming distance $2$ to $w'$, so the error cannot be corrected. The argument for identifying up to three errors is similar: at least four bits must be flipped to arrive at $w''$ from $w$ (as before), and anything less than that will not yield a valid code word.