Given that bones require calcium and iron is used in red blood cells, how are these materials gathered by the developing organism? Is it possible that very small quantities used as "references" are somehow stored in the human egg? Or is there some way of encoding for an element in DNA without any amount of the element already available?
No. DNA is not a blueprint, it is an integral part of the machine. There aren't little people in your cells who can read an instruction set. DNA, together with a living cell, produces (actually is used to produce RNA, which goes into producing) specific proteins, some of which bind to e.g. iron ions and integrate them into the enzyme mechanisms. It's all about making the right protein (e.g. hemoglobin to bind iron) in the right place and time. A living cell is a reactor, and the genes are your catalyst.
What kind of proteins your DNA produces depends on the surroundings. If in a skin cell, it produces proteins that directly or indirectly are necessary to produce skin. If (in an embryo) close to the surface, but not yet a skin cell, it produces proteins that later help turn it into a skin cell. And so on.
If an alien got hold of just our DNA and RNA, he could never ever even find out how a human works or what he looks like.
Naked DNA doesn't really code or do anything. DNA inside a living organism provides instructions that enable the organism to build things such as proteins which then do the work. In that context DNA also provides instructions to signal which things get made (some of the "code" tells the organism which things get made in which contexts).
As for how elements used in living things are "encoded" they are not in any meaningful way in the DNA. But proteins and other products made by cells have very, very specific functions. Some of those functions are designed to carry, transport or exploit the properties of elements. In plants, for example, a variety proteins in some cells are designed to assemble chlorophyll, the molecule at the heart of photosynthesis. Chlorophyll contains a haem group designed to bind to a magnesium ion and form the heart of a very complex system embedded in other proteins and plant membranes (all built from other cell machinery). The overall system enables plants to use sunlight to produce sugars from the carbon dioxide in the air.
But the ability of the plant to use magnesium derives from the specific structure of the haem in chlorophyll. This is magnesium specific the same way that much simpler chemical ligands built by chemists can be very element-specific (eg crown ethers and their congeners can be designed to very specifically bind alkali metal ions). But this specificity is serial steps away from the "coding" in DNA. The steps might be DNA codes for a bundle of proteins, which, in turn, build other proteins or other molecules, which in turn make up the machinery to make yet more molecules which are assembled by other proteins and molecular machines to create complex systems like the one used in photosynthesis.
The ability of organisms to exploit specific elements is based on the ability of parts of those final machines to very specifically bind particular elements in particular ways (even chemists know how to do that). But the machinery that achieves those goals is a number of steps removed from DNA which doesn't need to encode anything about those elements.