I also have a six year old who can throw a mean tantrum, it was particularly bad this year after she changed schools, with lots of face-pulling, sulkiness, paddies and so on.
I don't have a magic answer, but it is usually a phase, I found it just seemed to pass. I would stick with one or two discipline methods, like always losing computer time or whatever so it's pretty boring and predictable. The 'consequences' approach also works well at this age if they are clever (you see I think the naughty step is a bit young really, I just send her to her room or another room for a bit of quiet time, it's less a power struggle then and they still calm down). So, if she's not ready for the day, you can't go out and do that fun activity. If she's rude to you, you will be too upset to read the bedtime story. Stay firm, but I wouldn't crowd her or back her into a corner, let her have some time out, and come forward to you when she's prepared to be nice/not rude/eat whatever/be co-operative.
As for her laughing when you cried, she just didn't know what to do and so did something inappropriate. She's probably actually quite scared/confused by this response. I don't think it's a big deal, just tell her that even mummies cry and that although you do love her so much, her behaviour is not great.
Two other things that have helped us: ask her what's wrong at a quiet time. If she's clever then at 6 she can be starting to talk to you about her feelings. I found out my daughter was upset by a school situation with friends that I didn't know about and by other worries, some of which were quite absurd to me, but important to her. Then she'll feel like you listen (my daughter also wrote out sentences 'Today, I felt...because...) as she loved writing and found it easier than talking.
I also think it's easy to forget that at 6 they are still little, have emotional swings and still need lots of cuddles. I would do more cuddling, even if you feel a bit cross with her, I have noticed my six year old who seems to be so independent and mouthy is like a little baby if you stop and really give her a twenty minute cuddle on the sofa.
I do feel for you, I have so been there! Ignore any of my advice, it might or might not work for you, but you are certainly not on your own and your daughter won't be like this forever.