Are magnetic fields additive? When considering the field around a permanent magnet or current carrying wire, would it be accurate to say the magnetic field effects of each element add linearly in space, or is the interaction necessarily more complex?
How do materials like iron, with no net magnetic field, interact with the field surrounding actively magnetic objects? Do they change the field or are they simply acted upon by the field?
 A: You need to distinguish magnetic field from magnetic flux density here. These names are very unfortunate, because the full physical object is the flux density, while the magnetic field is just the external contribution. The magnetisation of the material is the other part. This is important when you consider magnetisable material, like iron in your question. In the absence of such material, like in air, field is the same as flux density. What is really not helpful at all is that sometimes people just use the word 'field' to refer to the flux density.
With regard to the field, as opposed to flux density: Yes, the values are additive. The flux density behaves different, if magnetisable materials are around. For instance, bringing two magnets close to each other produces a flux density that is not exactly the two (separate) flux densities added, because the two magnet's magnetization change as the result of seeing the other one's field. The amount of change depends on the type of magnets and the field strength they see of each other. Rare earth magnets, that are ubiquitous nowadays, have a very large coercive force, which means they need to see very strong fields for their magnetisation to change substantially.
With electromagnets you have a similar effect, if you use a magnetisable material for a core. For instance one coil can produce a significant flux density, but if a second coil drives the core material into saturation, the first will contribute much less than expected. For core-less coils this problem does not exist, the flux densities add up, as long as the fields are static.
With varying fields you can have a host of further effects. For instance with coils you have induction, which can also have a similar effect, depending on the impedance of the coil driver circuit. Magnets in a varying field may develop eddy currents in conductive parts (rare earth magnets are always packaged in a metal casing, as they are very brittle), which can also spoil the additivity of the flux density.
Material like iron gets magnetised in an external field, and thus produces flux density by it's own. It is this flux density that interacts with the external field, and causes an attractive force. The magnetisation of iron - being a so-called soft-magnetic material - changes as the external field changes, such that the force is always attractive, even if the external field swaps the direction.
A: Yes magnetic fields are additive. We call systems with this property linear systems and they can be studien in Linear Algebra. This includes all vector fields you may be familiar with.
Edit: The above answer is correct for fields in a vaccuum. If you start to add materials like iron the case becomes nore intersting. A way to think about iron is with each "atom"( may be a grouo of atoms called a magnetic domain) as its own magnet. (This is generated by a property of Iron where all its electrons in its half full shell each spin in the same direction.)
Each of these magnetic domain are in random directions. When you apply a magnetuc field, each of the magnetic domains line up and add in a linear way as described above. The stronger the field the higher percent line up. The non linear parts come in when all the domains are lined up and therfore you can't cause any more to one up even by increasing the overall magnetic field. In this case Non linear effects start appearing and the fields no longer add up simply.
This effect is called Saturation
