Hi J.-
I've worked with 3 year olds for more than 16 years and today I coach parents about working through issues like you have. I never had "terrible 2's" it was always between 3 & 4 when the issues popped up. As you can tell, your daughter has it the stage where she realizes she has control (over you) and it testing her limits.
Here is what I suggest:
When she demands you to do something, acknowledge what she said, by replying, "I hear you want something, but I can only hear nice works in a nice tone, like 'Mommy will you please..." This provides her with information about how you expect to be talked to and that you won't reply to demands.
Next, when she gets upset and demands again, I suggest you reply in a similar way, "I hear you're not happy with that result. When you find your nice quiet voice, please let me know what you were hoping for. I'll be ready to listen when you're ready to talk nice." If you can see what she wanted, then try "I think you wanted the pink cup and not the green one, if you'd like the pink cup, you need to say,'Mommy may I have the pink cup please"
When you provide your child with clear direction and limits, they will handle issues easier and test boundaries less. The key is to be consistant. Don't give in to the yelling, or she'll just fight you.
This is a stage of developement, just like when she was learning to walk, she tested boundries to figure out what worked. If you remember it's a stage of development she needs to pass through and learn from, it will help you deal when she's driving you crazy.
I hope this helps-
R. Magby