Hate the Japanese trope of become wet, instant cause for concern of catching a cold. Not how that works. It's stupid and overplayed.
Second, I can kind of understand it, but that looked so cheap. Maya comes in, sees Izuki wearing his jersey, and then Izuki stutters and fails to explain that she just got doused in water and that Okita loaned her his jacket. Yes, I can understand the stuttering because s/he's flustered. It's not unreasonable. It's just convenient for the author.
Third, yup, it was obvious she wouldn't say it yet. It'll take probably thirty more chapters for Okita to get a hint that Izuki is the same as Itsuki, and then HE will be the one to piece it together, not Izuki/Itsuki telling him. So it is predicted, so shall it be.