1. I would not use "cuts". We say "His lips were chapped".
1. Can I say "When it's really dry and windy in winter, there are some "cuts" in his lips"?
2. Can I say "You need to put a band-aid over your cut", as I think "wound" is way too serious?
Here, we might say his lip was split as a result of the cold. Chapped lips are less severe than split ones. And we have these every winter. For 6 months or so.