I suggest it's possible that she feels challenged by you when you say no. I also wonder if you give in to her eventually when she pushes. Have you tried giving her more choices and using different words than no. Perhaps say something like, "I know you want milk. You can have a glass with dinner." and then completely ignore her efforts to get you to change your mind.
Being obsessive about emptying the soap dispenser and snapping crayons in half is normal. When you see her doing this instead of saying not to break the crayons, pick them up and put them away. Put the soap dispenser out of her reach. It's too great a temptation for her.
As to her blanket, always have it available. Wanting it always, is the way a 3 yo thinks. Having her blanket helps her to feel secure. Sounds like, for her, not having it is quite upsetting. Do what you can to help her.
It really does help to anticipate problems and do what you can to prevent them. It also helps greatly to not challenge a child with words but to make it easier for her to choose the right thing. And it's necessary to be consistent in doing this.
Just having something unpleasant happen does not work at this age. Their brain is still very immature and they're not able to connect unpleasant with their activity unless what is unpleasant is a part of the activity. Always taking the crayons away, even not allowing her to have them for the rest of the day will make more sense to her than getting in trouble and having Mom yell at her in frustration will.
There are several books written about how to manage a strong willed child. I think one is even titled Parenting the Strong Willed Child. They will have additional suggestions. They will also help you to understand possible reasons for this behavior and thus be better able to respond in more effective ways.
After your SWH I suggest that you get an evaluation. Perhaps take her to a developmental pediatrician who can see what she needs and then arrange for treatment from the needed professionals. It does sound like she has some important issues that need attention. Refusing to see her father on Skype, etc. indicates to me that she is angry that he's gone. There is more going on than anger/depression but that is a place to start.
Eating non-food items can also be an indication of an emotional difficulty.
I would insist that the pediatrician listen to me. I suggest you make an appointment to talk with him/her without your child present. Ask for a referral to a developmental pediatrician. If your insurance doesn't require a referral then call your insurance and get the names of ones that they cover. Then you wouldn't need to see the pediatrician first.