Kooky vs. eccentric vs. crazy is both subjective and kind of a fine distinction, but I'm sticking with "crazy." Alyssa's being nice--both as is her personality, and because Riley is a family friend. Though that raises the question--which presumably will be answered in future updates--about how she and Scarlett got together.
Again, "crazy" isn't a negative here. Nova in Eternum is crazy, and I love her (she's a hell of a lot of fun). IMO, Riley goes beyond "eccentric." And she uses that word herself: "Eddie, tell this crazy bitch to get off me!"
So she didn't mean it? I don't think I'm buying that.
That's an interesting view of it. The only significant interaction I remember between the two of them is her corrupting Alyssa.
We don't really know anything about the "organization," and there can't be many people who'd sign up for "fighting across time." I'd argue that, if you didn't experience it yourself (which Riley now has, but it's pretty unlikely she had when she started doing whatever she does for them), you'd have to be kind of crazy to believe in the mission in the first place.
Likewise, "crazy" in this context doesn't mean "mentally unstable."