I think this is your big clue:
"I also think she asks for things just to be upset. For example, she will sometimes alternatively ask for something that I say yes to, like for food or to snuggle, but she'll run away from me screaming and not accept whatever she asked for."
In other words, she WANTS to feel upset and whatever you give her won't make her happy at these moments. That means she just needs to feel that upset.
Here's why. Human emotion arises and passes away constantly -- but only if we feel it, bring awareness to it. To whatever degree we "stuff" it or fend it off, that emotion doesn't dissipate. It is stored in the body. Or, as I like to say, your child stores it in an emotional "backpack" and carries it around. But those feelings want to come up and out -- that's a natural healing process. So the feelings bubble up, especially when the child feels safe, or when her conscious guard is down (in the middle of the night, or when she's tired.) But they scare children because they don't feel good. That's why the feelings got stuffed to begin with. So the child screams and cries and struggles and runs from the feelings. And from you, since it is your safety that is making the feelings come up.
I would say that your toddler has a full emotional backpack and needs to be allowed to do some emoting. I know it seems like she is already doing that, but since you are (naturally) responding as if it's an emergency, your toddler is feeling those emotions as even more of an emergency than she would already. So it's a challenge for most parents, but if you can stay calm and welcome your toddler's upset with as much compassion as you can muster, she'll feel safer. She'll even get beyond the initial screaming and running, and she'll cry and writhe and let the fear out. (My guess from your description is that she has some big fear stored up.)
Staying with her is important. It's ok to hold her gently, since many kids need to struggle to get the fear out. But if that doesn't feel right to you or triggers you to be anything less than gentle, just to stay next to her. You will need to "block off" her escape avenues by sitting in front of the door most likely. If she tries to hurt herself, hold her so she can't, and tell her that you will keep her safe.
What about nursing? While I would always nurse a baby under a year old who was upset, I think your instinct NOT to nurse at these times is right on target. Nursing releases soothing biochemicals. That's good when you're trying to soothe. But in these cases you are NOT trying to soothe. Your daughter is showing you that she has some big feelings she needs help with. Soothing her sends those feelings back into the emotional backpack. What she needs is to get them out. So when she gets like this, what she needs is your help to "show" you all those big feelings. Tell her "I'm right here...You're safe." Be her emotional witness.
Can you sometimes go ahead and nurse her, if you just aren't up for a big meltdown? Of course. The feelings will stay in there and come up another time. During the night when you aren't at your best, I would advise you just to nurse her if you can. During the day, go for the emotional release instead of nursing.
How long will this go on? She might cry or scream for an hour, and then fall asleep in your arms. She might repeat this daily or twice a day for a month. But every incident will diminish the weight of the backpack and will make her happier and more cooperative in between meltdowns, and eventually, she will empty the backpack. Then her meltdowns will diminish in ferocity and frequency. Because she has become comfortable with her emotions, she will "feel" them as they arise, and let them go, rather than stuffing them. That means no need for meltdowns, and no more screaming.
Where did these feelings come from? For some little ones, the feelings come from trauma -- a traumatic birth, separation, medical intervention. But it doesn't need to be trauma. Toddlers have big feelings. She may just be super-sensitive and have a lot of emotional reaction to things. It doesn't really matter if you know where the feelings come from, as long as you empathize with her so she can feel them and let them evaporate.
Should you get a crib? I think that might just make her feel less safe and more alone with the feelings because you can't hold her, and she might hurl herself out of it. Your best strategy here is connection so she feels safer.
Here's a link with more support about how to do this: http://www.ahaparenting.com/_blog/Parenting_Blog/post/What_about_those_days_when_he%27s_hell-bent_on_misbehaving/
Good luck, and please let me know how it goes!