The little hat on the vowel means that the Latin root word had an 's' there.If I remember, then yes.
Yeah I realised that it was that and as such hospital is different in that it's not spelled the same way. The h in hôpital is silent right? I'm not sure I pronounce the ô right though. And I have said hôpital loads of times btw due to L'Hôpital's rule.
H is always silent in French.