My dd did! Straight after birth, more or less. Then for about a month, on and off. We used some really tiny mitts (she was very small when born), and some slightly bigger ones when she grew. We did need quite a few pairs, as they got dirty through her sucking them, and got lost easily, either out and about or in the laundry. I would have a pair to hand in your labour bag, just in case. If money's tight, Tescos and such do plain pairs for about a pound.
Yes they scratch (themselves and you!). Little nails are surprisingly sharp and awkward to trim! DC1 has mitts but with DC1 a midwife said they now advise against them- I'm a bit sketchy on the reason but I think it was because the prevent self-settling?