Hi, about Kate and MC:
Kate and MC are, in love. The problem with them, is that the feeling is hiding/confused by brother/sisterly love. It's been there since so long, that they confuse it with "normal sibling love" even if they are friends, they yet didn't realize that, they really love each other in the normal way. I'd say they go beyond that love. Yeah, I know it's not the normal relationship story, I really tried to make something different for them, a double side of the coin love story with a slowly pacing.
About the identity and lies: They both had agree that for the sake of keeping each other safe, they'd never tell that truth, they both understand the dangers of what they're doing, yet, would never put the other in danger for their decisions. Of course eventually all this will blow up and they will find out, I believe we're not that far, meanwhile Kate will have to struggle with being in love with two identities.
MC can't control time or anything, it's pretty clear, when you lose or die in random fights you'll get the death game over.
In regards of the "save the girl": Yes, MC knew where the attack was, but Winter's info changed everything, he learn that it was a small attack, and trusting his abilities, he was sure of being able to take down 3 Hingetus, that's why Lynx team and Erick were close, but "not that close".
And he did good, even if it's unknown, them being there, saved a lot of lifes. I thought it was pretty well stated, but I'd have to take a look at the script to check again.