A carrier would have a couple launcher pods of RAM missles plus 3 more Phalanx gun pods. Ya gotta hope the screening force would thin the herd a bit before the carrier would need to worry about them. :)
I would hope so, but I have no doubt that Iran is able to muster a virtually unlimited number of suicidal conscripts. Better safe than sorry.
The traffic lane in the SOH is narrow enough that a disguised vessel(s) outside the lanes but inside Iranian waters armed with a hidden Russian made gatlin gun could within a few seconds preemptively blind the radar sets of every port, or starboard looking SeaRAM and CIWS on a carrier.