Sounds like you are both having a very shitty day. Can you start over.
Three year olds are barely human!
It's very tough when you are left on your own with them, I have two and gawd they can be monsters. From time to time I just feel like screaming and putting the whole bloody lot on ebay as a deterrent to teen sex.
Go have a cup of coffee, sit at the table so it can't be tipped on you, show the girl a pot of glitter and a piece of paper and you will at least get ten minutes peace (even if you look like Julian Clary's armpit for the next week as a result!)
I can empathise with you. Having similar trouble with DD, although thanfully right now she is calm (enough). I wish I could offer advice, but instead I am going to watch this thread for advice too. I hope your day gets easier.
My advice is to not allow her to do these things. You are the adult, she is the child. Physically remove her from situations and take her away to time out for three minutes. When she's climbing on you, stop her, physically. You need to show her by actually doing it because to reason with a child of this age is extremely difficult and could end up in a battle of wills.
Take her to time out for three mins, explain why she's there,and leave the room. If she gets up, put her back. Again if necessary, as many times as it takes. She will soon see that you are not going to give up. After all, YOU are the one in control. I'm sure I'll get flamed by people who are pro unconditional parenting and that I should listen to my child about why she's throwing yogurt on the floor or having tantrums. Of course you have to listen, but not to discipline IMO is just letting them learn bad habits. Being consistent is key, don't give up.