Hi D.,
My best advice is be patient, calm and prepared to repeat yourself a lot! My daughter started the same behaviors at about 16 months, but I've found that she responds pretty well to the question "are you supposed to do that?" She'll say no and keep doing it anyway, but then if I suggest that she sit down or do something else, she will. Then two seconds later she'll stand up again and we start over, but I figure she is learning what the right behavior is. When I get tired of the situation or she starts to get out of control, I just get her down from the chair or change the situation, whatever it is. Keeping a calm, level tone and giving desired alternatives seem to work for us. And lots of praise for doing the right thing!
In terms of hitting and biting, we went through that a few times too just recently (she's almost 20 months old now). We don't do time outs per se, but I do leave the room if I am too frustrated - a time out for me! After a few times where she hit me on the head, I told her that next time mommy wasn't going to play anymore if she did that, and I went out of the room. Well, she did and I did and she screamed and cried a lot when I left the room - but when I came back in a few minutes later, she did it again. I told her that was that, no more hitting, and put her to bed (luckily, it was close to her bedtime already). She cried, but when she woke up in the morning she was fine, and then we had a talk about how much nicer petting is then hitting. So now she "pets" mommy or daddy or the cat or dog when she has that urge to hit - I can see the impulse in her eyes and watch her as she stops herself and chooses the action the has better consequences.
Same with biting. She bit me exactly twice, and I did the same thing, and she's never bitten since.
Keep in mind that they don't mean to be doing something wrong or bad - they are just exploring their environment and their limits, it's their job. And your job is to help them find those limits. In my opinion, that works best with a positive rather than a punishing approach.
good luck!
best
K.