Based on the conversation Angela was having with Whiterose (the leader of the dark army), I think I know why Angela and Elliot are so important in the story of the show. It's why they are in the middle of everything in this global meltdown. And its why their behavior, above everyone else's, seems so erratic.
First of all, both Elliot and Angela work for the Dark Army. That's not the crazy part. Hear me out.
When Angela and Elliot were young, they were entered in a program where people can have two personalities implanted in them (this could be during the radiation disaster that Angela is obsessed with potentially). Anyway, the procedure made them have something along the lines of mental super powers and with the change they can achieve more in their lives though the use of multiple personalities. Their brains are like partioned hard drives. Why does Elliot always seem to disappear within the story? He's working as Mr. Robot on an overarching plan. Back when they were young, Elliot was implanted with his Dads personality, and Angela was given her mother's. This is why he's so great at hacking - two brains are better than one. This is also why Angela is so successful at a young age at E-Corp. However, Angela's role is a little more complicated...
This theory also explains the questions Angela was asked in the room where she was kidnapped into. The Dark army was testing who was in control of Angela at the time. Red or purple? Giraffe or seagull? Do you cry during sex (AKA "does your other personality ever come out in stressful situations?"). This explains why her behavior is different after she leaves. She switches into her other personality - due to something the Dark Army does to her off screen. Elliot's dad died willingly to have his personality fused with Elliot, and Angela's was killed to fuse with her. Now they work as super powered agents in order to bring down capitalism/America. This is also why she is able to get so close to the E-Corp CEO - she just doesn't know it. Angela is actually a double agent for DA at E-Corp. In this theory however, the CEO of E-Corp could potentially know it, which explains why he keeps her so close. Remember the beginning of the last episode? The two rooms where he is not the most powerful person in the room are rooms with either Elliot and Angela in them.
this part is a bit shaky, but this could also explain why the leader of the Dark Army cross dresses: she switches personalities to get things done. When she's asking Angela about how she is in the middle of everything, it's because she wants to unlock her own full potential. That's also why she is so big on staying on time - she can switch into different personalities and knows it, and thus uses her time to the fullest. The Washington Township disaster could have also been her project at E-Corp which links her to them as well if that was actually her doing. The timeline matches up.
Back to Elliot, Stage 2 is where Tyrell's personality also joins into Elliot. That's why he died and Elliot can't remember. He was assimilated into Elliot by the DA and now Elliot has 3 brains/personalities/sets of intelligence to do the work for the DA. Elliot and Angela are first of a line of new age super soldiers, perfectly fit for the age of information, and the Dark Army has control of them. This episode talks about lucid dreaming, because this is how Elliot will harness all three personalities at once. Tyrell will now help Elliot with stage two of the world takeover plan that the Dark Army is trying to conduct.
BOOM.