J., you're expecting your 2 year old to act like a 5 year old. She can't do it.
You need to read two things. One is about child development, specifically this age set. Another is The Toddler Creed. Google it and enjoy. (Yes, really, I meant enjoy.) It is all about your daughter and every other 2 year old.
About the food, please don't tell her that the food isn't hers and that she has to say please in order to eat. If you are willing to let her have something, then don't put "please" as a condition for it. Food is a right, not a privilege. If you have food out that she is not supposed to have, then put it away. If you are eating when she can't eat, don't eat in front of her. You're expecting her to understand that it's not meal time or that you can eat what she can't. She's too young for that and you shouldn't expect for her to understand. And the last thing you should do is tell her "no" for stuff and then let her do it. You're going to be really sorry if you keep that up - she will learn to manipulate you by having meltdowns and tantrums to get her way.
She IS going to say "mine!" for everything. Stop fighting with her over it and just continue to teach the sharing thing. Don't expect her to learn it for several years. Teach her to say "thank you". That's what you want to hear, especially for food. You can teach "Please pass the potatoes. Thank you!" rather than withholding from her until she says it. Or you can say "What's the magic word?" and SMILE at her rather than make it a tug-of-war. Say the word please FOR HER if she doesn't say it. That way you are teaching rather than demanding.
This is the time that tantrums and whining start. When she starts to whine, don't say no to what she says. Look at her ONCE and say "I can't understand you when you whine. You need to talk to me in your nice voice so I can understand." Then you turn away and ignore her until she uses her regular voice. As soon as she does, you "turn back on", look at her and deal with her right then and there. And you don't say NO, and then say YES. You show consistency and stick to your decisions.
That means think before you tell your daughter no. Try to say yes to as much as you can! Yes can have caveats. Yes, you can go outside, but first we must do "x". Yes, you can see Dora, but first we must "x". You already had Dora time, so now it's time to do "x". The more you can take "no" out of your vocabulary with her, the better things will be.
When she has a meltdown, say to her "you must be tired, so you need to go to your room" and put her in her room and close the door. Don't let her come out and don't let her know you are outside the door holding on to the doorknob. She will finally stop having her tantrum because she has NO audience. When she is quiet, you "show up" and open the door and say "Are you ready to talk nicely to me?" Then tell her that we do not yell, hit, jump up & down, whatever she did when mommy says she cannot do something. Tell her you know she is disappointed and sad, mad, whatever you want to call it. It's good to give her emotion a name and acknowledge it. However, she will have to go into her room alone if she has a tantrum.
If you do this EVERY SINGLE TIME she starts having a meltdown or tantrum, she will make the connection that her tantrums get her NOTHING. She will grow into stopping having them, unless she has special needs and cannot control herself. Right now, she WILL have trouble controlling herself because she is too young to get your lesson until she has been put in her room over and over. And with the whining, the same applies.
You must be 100% consistent. And you must be patient. Understanding where she is developmentally is key in accepting your daughter for who she is and helping shape her behavior over the coming developmental stage.
Good luck,
Dawn