It's not so much about ignoring the hitting or biting, as not rewarding her for such behaviour. So if she hits or bites another child, ignore her but make a huge, huge fuss of the other child. If she bits you, say "No" in your sternest voice, then walk away from her - preferably into another room where she can't see you. Two minutes is enough.
If she tantrums, walk away from her and leave her to get on with it on her own. Once she's stopped, go back to her. If she starts again, walk away again. When the tantrum stops and she's behaving normally again, give her a big hug.
If you're out then the same rule applies - turn your face away from her, make a fuss of the other child. If she hurls herself onto the floor in the supermarket, walk a little way away - somewhere where you can still see her, but she is aware that you have gone. Don't make eye contact with her. And ignore all the tutting old biddies who are bound to roll their eyes and make comments.
WRT behaviour when you're out somewhere like the Snow Dome give her one warning - stop that or we will have to leave. And if she carries on then leave. Carry her out if need be - don't speak to her, don't look at her, just remove her from wherever you are. You can ignore her whilst you're out though; just turning your face away from her and making a big fuss of whoever she's hurt (even if it is your dh!).
You're right about wanting the drama of it all; she likes attention - any kind at all, even negative - and she has found the perfect way to gain your attention. Which is why ignoring/turning your face away/walking away can work for some children.