I'd settle for the small lag with a humidity sensor. If you're the type to run the shower for a minute before getting in, it should kick in in that time.
Could do a presence sensor or motion sensor pointed at the shower. Haven't played with either, and you'd have to figure out water resistance.
I just got Easy!Appointments setup after being disappointed by cal.com. I don't see any options for ics, but it does have CalDAV integrations (along with Google sync as well). I haven't personally used the CalDAV integration so can't speak to how mature it is. I've got my SMTP settings setup and emails go right out