Is it normal for babies to sweat?

Yes. Sweating is the only way they have of getting rid of excess heat they produce by metabolism.
Yes. Some babies sweat a lot more than others. Younger babies are often overdressed, so they work up a sweat during feedings. Young babies also can't kick their blankets off, so sometimes they sweat a lot during sleep. If a baby seems to change colors (turn bluish, dusky, or pale), has trouble breathing (breathing very fast when not doing much), or has other odd symptoms, a doctor should evaluate.