My son (now 8) went through a picky eating state early on. Our strategy was to not make a fuss about it, but also to not cave in and serve special stuff. We made dinner, which may have included items he "didn't like" or refused to eat. Any whining was mostly dealt with by saying that this is what's for dinner, it's the food on your plate, eat it or don't eat it, but there certainly doesn't need to be any freaking out about it. We didn't make any special food just for him, and other than a rule of trying one bite of everything served, if he didn't eat a certain thing on his plate, that was fine.
However, it is important (I think) to establish the rule of try one bite of everything. That's a rule to this day. There are certain foods that my son legitimately doesn't like (green peppers, for example). He tried them several times, and it's clear it's a taste thing and not any phase or stubbornness, so we respect his dislike and don't serve him that food. But it takes awhile to differentiate between a phase and true dislike, as kids' tastes change as they develop. At 4, he decided he didn't like potatoes. It lasted about a year, he kept trying one bite any time there was a potato in front of him, and then one day realized that he liked potatoes. He also decided for awhile that he didn't like "soup" though if we served him soup and called it "stew" or "noodles in sauce" he would gobble it up.
If you're concerned about your daughter's weight, I would definitely take her in to her pediatrician and make sure there's not a medical cause. The reality, I find, is that kids will eat when they're hungry. Maybe she tries to push boundaries for awhile to see how much she can play on your mommy guilt and worry, but eventually, if chicken and broccoli is what's for dinner, she'll be eating chicken and broccoli.
And really? Any melt-downs should be sent away from the table, don't you think? Emotions are great and I encourage them, but the family table isn't the place. If she's upset, have her get down and go to her room - not as a punishment, but just as a "hey sweetie, it's okay to be upset, but you need to take it to your room. When you're feeling better, come on back".
I believe that kids are just little bundles of feelings and thoughts and opinions and emotions and they're learning every day to sort through them and figure out who they are. It's our job as parents to provide boundaries, so they feel safe and can clearly see where those limits are and work on keeping themselves within them. Consistency and maintaining your own even, understanding tone is the key! :)