If I die as a regular iron man and I lose my iron man armour, will I be able to retrieve my armour in Lumbridge from Adam? Is there another retrieval mechanic? Or perhaps the armour is lost forever?
If you look at the "Destroy" description of the platebody for example on the Wiki, it states:
You can obtain another platebody from Adam, Paul or Juan in Lumbridge.
The same is said for the rest of the pieces of armor. Thus if the item is lost in some way, you should be able to get it back. Now if you were a Hardcore and died, I assume you will not be able to get back the hardcore ironman armor, since after death you are demoted down to a regular ironman.