Sorry, cross posted with later examples.
Maybe for now, just stop putting her in a position where she has to share. Praise her brother for his good behaviour in front of her and try ignoring some of the "less bad" behaviour. Consequences that are appropriate for the very bad behaviour.
So as close to the event as possible and timed appropriately.
It really sounds like she wants to be autonomous and have a bit of independent control.
"Dd we have to leave in ten minutes. If you don't make a fuss, perhaps we could stop by the shop afterwards and you could choose some sweets or a magazine/choose a film for us to watch when we get home?"
Was there really an issue with her pushing the trolley? Was there a different battle you could have picked? Or if you were worried she'd bump into something, "DD, thanks for wanting to help, that's really grown up of you. I just think it's safer if I push the trolley. If I give you a list of things, would you run and get them for me or is there something else that you think you'd be able to do to help us get back home quickly so we can (insert fun activity here)."
Do you think she's just feeling a bit controlled and like you are always on at her, by any chance? ETA pretty much every example you've mentioned is a control issue...in her mind you are making her share/taking her sweets/forcing her to leave/not letting her push the trolley. She needs to feel like she has SOME control, ND or NT.