S.,
I also have a 3-year old who has lots of character, and some of my issues with him are similar to those you have with your daughter.
What I do when he's "acting out" (that means totally not obeying and just being generally naughty) is take him in my arms and contain him for a while -- I always find a place to sit. Sure, at first he'll scream and ask to be released, but I won't let go until he can calm down enough to control himself. I'll kiss him, and whisper in his ear, "no, I won't let go until you've calmed down" (might take 2-3 minutes, or more), then when he does, sure enough once I let go he turns to me for a long cuddling hug. At this point, while still hugging him, I can talk to him about what he's done wrong, or even wait until later. This depends, and is your choice. Your child will understand she's crossed the line.
Speaking of, let her know what the limitations are, and stick to them, no matter what, with the occassional compromise. Just be very clear, so she knows you mean business. For example, my son wants to hold a stick in the car, and the rule is that all sticks must be on the floor when we drive. He almost got me recently when he asked to hold just a small twig, after all, what's the harm in this? I decided to keep to the rules, so he knows I am serious about them. On the other hand, yesterday he wanted to jump down the stairs of his day care (he jumps 2-3 steps at a time while I hold his hand) and my compromise was to tell him, "no you must walk down normally, and at the last two steps you can jump." A child has to save face from time to time, right?
I do notice when he's tired he's much more awnry and naughty, so it's true regular sleep and consistent meal times will help. I hope she takes a mid-day nap, and even if she insists she won't, you insist the rules are that even if she doesn't sleep she must stay in her bed - with a 99% chance she will sleep. I've also decided to set time limits on his bedtime ritual. We can read for 15 minutes, then "goof off" for 10-15 max (tickles, hugs, blow kisses), but I will give him a 2-3 minute warning and if he wants me to be there to tuck him in, and give him a small back rub and a goodnight kiss, then he has to be down in bed and willing to stay there. It works! Or, if it doesn't, he gets one last chance. I will tell him that I'll leave if he doesn't cooperate and he can tuck himself in bed!
The more you set clear limitations, with clear, simple, instructions, the more your child will obey and listen. And, stay patient and as calm as possible, when she's naughty, as that's important both for her and you.
As for the time outs, I gave up on these, now that he's three, given that for him they really were not working. I just stop him in his behavior, get down to his level and say "no, I'm going to need to stop you until you can learn to stop yourself." (Terry Brazelton, pediatrician - do get his book on discipline which you can find on the internet) Then I can explain to him what he did wrong and try to get him to tell me in his own words after that. I find this helps for him to start taking responsibility over his actions.
As for not saying sorry, if I ask my son to do so and he won't, I just don't insist. He knows he's suppose to, and may just not feel like it. Then again, sometimes he really surprises me, saying sorry at the appropriate time without even being asked. So, don't put too much pressure on your daughter for this. It will come.
I am sure you'll get more good advice, but honestly, the containment in arms and the clear limitations has been a real life saver for me.
Good luck!