Between the 'loving to be told off' and wanting to be held/on your knee all the time, it sounds to me like she is desperate for more of your attention. That isn't meant as a criticism – some children are near enough bottomless pits when it comes to attention, however much you're already giving them, and negative attention is better than no (more) attention.
I think I'd try really looking for every little chance to praise good behaviour – try to squeeze in praise for that one sip of juice she takes nicely, the brief instant she's sitting on the sofa, etc. Try to get her 'helping' with tea and other jobs, and give loads of positive attention whilst doing it.
On the negative side, ignore what you can (personally, I'd leave the walking battle for now and just take the pram, for example) and for other stuff, stop her and say why, but as briefly and neutrally as possible, as if you're bored with it, or even as if you're a recorded voice message – I'd be trying to give her as little reaction and as little attention as possible. She stands/runs on the sofa, sit her on it, 'sit on the sofa', nothing else. And so on. For the common problem points, I find it easier to stay bored and neutral if I've planned my set response. I'd try to do this with the making herself sick at home, although I suspect I'd find it challenging!
Hitting and biting, do you have somewhere safe you can put her briefly? Behind a stair gate, in her cot, etc.? If so, can you put her there so you can physically withdraw from her (just for a minute or two) when she hits, as a natural consequence that people don't want to play with people who hit? If there's another adult around, even better if they could do it, coupled with the same neutral 'no hitting', and then give you lots of attention as the one hurt.
Finally, the getting so worked up out and about, do you think she is genuinely distressed by the different places and people? It's a lot to take in for our little ones as the world starts to open back up. Or do you think it's perhaps that when you are out, there are more things competing for your attention, and it's again trying to get mummy's attention any way she can?
Okay, that was long...hopefully I've managed to say something that was useful in all that text.