to think my 2 year old's behaviour is extreme?

Normal. Try distracting her if saying no doesn't work.

This too shall pass.

Actually I think that IS extreme. And I'm a mum and childminder so I've seen a fair few toddlers in my time. I'm not suggesting she should be 'diagnosed' with something but I can absolutely see why you are finding her behaviour difficult to manage. How is her sleep? Is she easier to manage when you're out and about?

They smile and laugh when being told off because they don't understand why you're cross/sad and they are trying to cheer you up. That is what the latest research says. It's not spite or ha, I win. You need to redirect them to activities you find acceptable so that they learn what they should do instead.

For posters saying it's normal behaviour. I think it is normal and DD has a lot of energy.

However, 2 year olds do have some self control and are aware what's right and wrong, and what kind of behaviour is unacceptable if you reinforce this.

What are the consequences when she pushes the baby?

Try saying her name in a firm and loud tone.

She should not be pushing the baby. If she tries to do this, turn her round quickly to face you and tell her clearly and firmly 'Do not push your sister'.

Oh yes I forgot about the little kids version of how to talk. Agree totally. OP I think you have the right idea but just unfortunately it's a bit early!

My ds is a nightmare, 2 and 4 months. He hits and bites. I was putting him on the 'thinking spot' but he just runs around and seems to quite enjoy it. He is deliberately destructive, has terrible tantrums a couple of times a day and frankly is bloody hard work. He smirks and laughs when he is being told he is hurting us etc.
He gets a warning now, it is a bit 'you can have the first punch but next time'..... he will deliberately to the bad behaviour again. I put him straight in his cot and shut the door for 5 minutes. Before he gets out I make sure he knows why he's there and then he apologises. It seems to help for a little while.
People keep saying he will grow out of it. When he does I'll paint over the scribbles on the walls and furniture.

Sounds normal to me. Using verbal instructions/reasoning at this age isn't really likely to help which is probably why you find yourself ending up shouting.

Control her environment rather than expecting her to control her own behaviour. So when the baby is in the highchair, she's strapped into a booster seat at the table too, physically pick her up and take her down from tables, use stair gates to block off dangers. It's hard work as you have to be supervising all the time, 2 under 3 must be really difficult.

The website Aha parenting has some good tips for two year olds. I think it comes down to keeping them busy so they don't have time to get into trouble, baby proofing and having age appropriate expectations. I found pre warning of what would happen next and how I wanted them to respond also helped.

That sounds tough OP, but FWIW I've known lots of equally 'spirited' toddlers that eventually mellowed out and had no SN (I used to teach pre-school)

What do you do when she does sonething dangerous? (eg pushing her sister?)
I tend to go with saying 'kind hands, gentle' and demonstrating the first time DS is a bit heavy handed, but if he does it again I pick him up and move him somewhere else and tell him why in a firmer voice.

Toddlers dont properly learn impulse control till later than we think...about age 3/4...so tbh if she's smirking when you tell her not to do something i'd probably keep picking her up and moving away.

You might want to read 'how to talk so little kids will listen' for some strategies. Sounds exhausting but normal to me!!

Sounds fairly normal to me. She sounds like she wants to play with you/get your attention. Could you small plastic boxes (empty ice cream tubes work well for that) with play do, small toys, etc... that you can give her to distract her when you are busy?

Normal.

Telling her not to do it or explaining why won’t work. At 2.5 they cannot understand reasoning and have very little self-control.

Having a younger sibling to compete with for your attention exascerbates her behaviour.

Distraction is the way to go. Remove her from the unwanted behaviour and get her doing something else which you can praise her for.

What do you do When she pushes the baby? What are the consequences?

Yes. Its down to you to develop strategies to cope with and manage her behaviour particularly when she is out. Its horrible but you will find something that works for you and you will get through it.

There’s no such thing as normal. Some kids are just more energetic than others. I had two very calm ones but then dc number 3 is like a whirlwind, she’s 8 now and still never sits still. Your dc sounds very ‘normal’