Got a few tiny cuts on 1 hand 2 days ago. No swelling, fever, drainage, etc, but there is a reddish spot near (not on) one cut. Could it be infected?

Could be. But it does not sound like it is. Any time you injure an area there may be some degree of an inflammatory reaction. This does not necessarily mean infection. If the area of redness increases, becomes more tender, warm to touch or swollen, see a doctor asap. The hand is very delicate and injured easily. Be careful and be safe. Hope this is better soon.