I agree with Molly's advice. Obviously you cannot let your child hit and bite others, but putting her in time out may just increase her frustration and make her act out more. I don't use time out. If DD hits or pushes, I attend to the other child and then DD is removed from the situation, but not punatively, and I tell her why and when she feels she is ready to rejoin she rejoins.
2 is still practically a baby. And now yours is suddenly displaced by a very needy baby. She is also probably picking up on your stress and exhaustion (which I imagine you have if you have recently returned to work). It's a big adjustment for everyone in the family. She is acting out and does need understanding and reassurance that she is not being replaced. This does not mean you will let her go about being violent and just talk about it. It means she needs more time and understanding from you so she can feel secure with her place in the family and she needs you to teach her how to act appropriately. I second the recommendation of the Harvey Karp book.
My DD is 2.5. She's recently gotten to a point where she can restrain herself from grabbing toys away from others. But before we got here she would tell me over and over about how she took the toy from whoever. She seemed pleased with herself when she was telling me. To me, it seemed that she was trying to process it, not that she was boasting her pride in her behavior. We'd discuss it whenever she'd bring it up, remind her of the rule. I always reminded her of the rule before being with other kids, etc. She got it. Your DD may be doing the same thing by seeming "proud" of hitting. She isn't proud of it, I don't think. She needs a lot of consistent reminders that that is not appropriate behavior. Discipline is about teaching, not punishment.
Though she may know something is wrong, by telling you sorry or time out, knowing it is wrong and being able to restrain herself from the behavior are completely different. She is still very young and still of an age that she needs a lot of help to control her impulses. So if you can recognize triggers (tiredness, craziness with the kids, hunger, whatever) then those should be removed to minimize the times when she will act out. You need to help her to act appropriately.
As re: the biting herself. I would ignore that. If it is just normal attention seeking behavior and you treat it like, oh we don't do that (but without alarm) then it should stop. I read one book where the guy talked about a toddler who was ripping her hair out. Every time she did the mother freaked out and tried to stop her. He said, the message being conveyed to the child was "this is really important" so the child continued to do it. However, if the mother had simply ignored, or where you can't igore and need to teach the rule then reminded without a lot of emotion, and then move on as if it didn't happen, then the child would move on before long. I have used that technique with a lot of undesirable behaviors. Different things work for different kids, obviously, but it may be worth a try.
I find that when DD is acting out the most, it is because she is getting the least attention from me and DH. So maybe try adding in some special time regularly with her (hard I know with a tiny baby!), maybe that will help address some of her needs for reassurance.