I've been there and done that..So I know how hard it is to think straight.
But, here goes...
First, you have to think of it from her point of view. You are much bigger than her. Adults are quite scarey when they shout and scream; she may be just defending herself or imitating you when she screams. If you shout at her a lot she probably doesn't know the difference between really bad behaviour and just irritating behaviour.
She is getting attention for screaming, just in the same way you expect her to give you attention when you scream at her. So you are reinforcing her pattern of bad behaviour, even when you think you are trying to enforce good behaviour.
She is only little, so her concentration and listening skills are not highly developed. Shops and restaurants at that age for some children are very stressful, even if for adults they seem like a treat. Some children are fidgets and don't sit still for long.
When children are fussy eaters their appetites can shut down - they genuinely don't feel hungry because the meal time is too stressful for them. Bad associations with mealtimes cause a kneejerk reaction to food. She also may not need as much food as you think. Could you feed her healthy snacks (little sandwiches, milk, bits of fruit) at scheduled points in the day to break the bad meal association? My dd was terrible at mealtimes; I know see it was a stress reaction to being told to eat things, and also an attention seeking device. She had a tantrum at most meals. I can't believe I let the situation get so out of hand...I should have sat with her and chatted, and stopped worrying what she ate or how she ate it, and I genuinely think I could have then defused the windups and the demands (which incidentally, never happened when she was at someone else's house!)
I could go on...but there is a wonderful book which I URGE you to read called
HOW TO TALK SO CHILDREN LISTEN AND LISTEN SO CHILDREN TALK. It doesn't mention naughty steps but it explains how to get children on your side so that they do what you want, and what they want, which deep down, believe it or not, is please their parents.
A bit of Supernanny which is completely overlooked is that she tries to "Involve "Children as well as putting them on the naughty step. Try finding good things to say to your little girl every hour, notice all the things she does well, even if it's just choosing a particular toy to play with, tell her in very specific ways what you like her doing, not what you dislike her doing.
Good luck, take a deep breath, you can turn things around. And 3 year olds are a total windup, it is not just you! But she is only 3 and you are a grownup, therefore reasonable 