Again...Christopher as pointed out before Alexis "acting out" is precisely what she is doing because for really the first time she is dealing with things she has never had to before. I don't think she really knows how to deal with them. Alexis has spent most of her life trying to be the perfect smart daughter. Now she is suddenly faced with things she can't control or deal with and is dealing with them in the manner that she has this season. Alexis has no focus. The rejection letter has probably affected her more than she has let on.
As for referring to Kate and Rick's relationship as "Caskett"...what's the harm in that? Refer to them as you will and I'll stick with Caskett
Finally Captain Gates had a very specific reason for working directly with Kate, she told her so herself. She wanted to see how Kate worked alone...without Castle or Ryan and Esposito. While a contrived plot device it still paid off in my opinion and it was an excuse to have her in the episode. So far they've used Penny pretty sporadically.