3 year olds can be testing.
But it might be worth considering your approach.
You can be strict and hold a strong line on behaviour without shouting and without being “sharp” with her.
If she cries everyday coming out of nursery think how you could avoid that. Or what the reason might be. Is she overwhelmed with tiredness, excitement to see Mummy, is she hungry or overstimulated?
Could you change the record if you picked her up at nursery gave her a big excited hug and said how much you’d missed her during the day? What if you said “I have a lovely snack for you in the car, can you walk really nicely” or if you made a game of doing silly walks to the car?
She came downstairs crying for “no reason” but it won’t have been no reason, it might just not have been a logical adult reason.
What if you had tried, what’s wrong? Where has my smiley girl gone? Do you need a big hug?
Bed time and bath time can be difficult transitions. Try giving her some choices, which bubbles shall we use, which PJs? Go and choose some bath toys. Go and pick a dress for tomorrow, or socks or a hair band or what ever. Sing a song, make it fun. Give her time warnings like “five more minutes and then lovely bath time’”
Strict doesn’t mean shouty. It doesn’t always mean punishments.
You shouldn’t give into tantrums but I never found a tantrum reduced by yelling. Better managed by saying “I’m not going to change my mind, but let’s have a hug and read a book/dance in the kitchen, you choose the music etc”
Hitting and pushing. Remind her it hurts other people, ask her how she would feel if someone hit her. Remind her her sister won’t want to play with someone who hurts her. Encourage empathy. And never ever allow hitting or push in to be rewarded with what ever she wants. And tell her that “I love you but I won’t fix you hair if you hit me” etc
She’s 3 she needs help managing her big emotions. Calmness, consistency and reason are more effective than shouting.