I believe that there is a good chance the idol was merely meant to distract from the fact that the room is a death trap.
The room maybe similar to the ending room/construct of Revelation Space, where the aliens have lured the humans to a particular place with an origin mystery simply to assist the aliens in eradicating humanity.
Once the atmosphere hits the room the bio-agents are set in motion, the only thing that needs to happen is for the explorers to stick around long enough to get infected. But it does seem like an awful lot of canisters just to form a simple trap, though apparently they have quite a stockpile overall.
Perhaps the Engineers that built the bio-weapons facility also had rivals or enemies that they sought to guard against. Rigging a central room with a booby trap, might be useful as a security measure.
On the other-hand in the intro, the aliens appear to have had at one time a reverence for life and creating life. By that same token, maybe they attach a religious significance to causing death. Apparently, Scott considered having the Engineers be Jesus at some point but felt it was a little too blunt for mainstream audiences.
From Alien vs Predator wiki - Engineers:
This explains the cave drawings 'inviting' humanity to track them
down. Some fans have suggested that Jesus Christ was an Engineer and
his death at the hands of man was the point they turned against
humanity. In an interview with Movies.com Ridley Scott acknowledges
that this back story was considered but neither confirms nor denies
that this was indeed used in the final script:
Movies.com: "We have heard that it was scripted that the Engineers
were targeting our planet for destruction because we had crucified one
of their representatives, and that Jesus Christ might have been an
alien. Was this ever considered?"
Ridley Scott: "We definitely did, and then we thought it was a little
too on the nose. But if you look at it as an 'our children misbehaving
down there' scenario, there are moments where it looks like we've gone
out of control, running around in armor and skirts, which of course
would be the Roman Empire. And they were given a long run. A thousand
years before their disintegration actually started to happen. And you
can say, 'Let's send one more of our emissaries to see if he can stop
it'. Guess what? They crucified him."
Either way the idol was probably built to inspire awe, but it's not clear whether the idol is intended for humans or Engineers, or whether it was built with a trap in mind.
What is the giant head for?
A trap and/or to inspire awe, religious or otherwise.
Did every ship carry one?
I don't think they were on the ship yet when they found the room.
Do they worship themselves?
As Orac would say this question is illogical since all religions possess some elements of self worship. But kidding aside, I would say yes in a sense, though as a god-form or a mortal I cannot say.
Why the mutable patterns on the ceiling?
Probably decorative art, whether murderous or religious I'm sure the Engineers felt they had to have a sense of style.
Another theory: The idol and room were for racial reunion, but that idea was abandoned in favor of a death trap:
What if the room was originally intended to be for a species reunion for the two species Engineer and man. But after some negative interaction with humanity the idea was abandoned and then the room was booby trapped when some of the Engineers started to have second thoughts about creating other intelligent species.