Is the total number of matches odd or even? How many times do you need to ask at least to determine the odd or even nature of the number of matches in box No. 1?

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Ask 3 times. The first time ask about boxes $1-15$, the second time ask about boxes 1 to 8 and 16 to 22, and the third time ask about box 1 and boxes 9 to 22. If the number of matches in the 1st box is odd, there will be 1 or 3 odd answers in the three responses; otherwise, it is even.

In fact, adding up the totals of the 3 times, in the resulting sum, the 1st box appears 3 times, and the other boxes appear twice or not at all. Therefore, the parity of the sum is the same as the parity of the number of matches in the 1st box. Asking only once or twice is not enough. Because in the two questions, 1st box must appear exactly once. Suppose the 1st box appears in the first question, along with box $i$. If the 2nd question does not contain either the 1st box or the $i$th box, then changing the parity of both these boxes simultaneously does not affect the answer. If the 2nd question contains the $i$th box, since the 1st box does not appear, there must be some $j$th box that appears only in the 2nd question. At this time, changing the parity of the 1st, $i$th, and $j$th boxes simultaneously does not affect the answer. Therefore, it is impossible to determine the parity of the number of matches in the 1st box.