Is spin necessary for electromagnetism? I know that spin is needed for defining the magnetic moment of any particle, and I have also read that the spin actually is the reason why some materials are magnetic. What I want to know is whether spin is necessary for the some interactions in the electromagnetic field.
Let me expound a bit: in classical electromagnetic field theory, the electric and the magnetic fields could be considered as some combinations of partial derivatives of the vector potential ($A_\mu$). Any particle couples with the field and interacts with other particles through it.
Moving on, if we consider the quantum field theory version, we have two particles coupled with the electromagnetic field, which then interact with the exchange of bosons (photons). My question is: how big of a role does spin play in the interactions which happen through the electromagnetic field? Are there some interactions which spinless particles cannot have, but those with spin can?
 A: Classical electrodynamics is formulated in terms of macroscopic (i.e. averaged over many atoms/particles) fields and sources (currents and charges). It is fully condensed in Maxwell equations and supporting material equations (describing how the sources respond to the fields). As such it does not need spin, simply because it doesn't care about the origin of the magnetic moments involved. Indeed, interpreting spin in macroscopic terms, as a current due to the particle rotation, is known to be incorrect: quantitatively for the charged particles, and qualitatively for the neutral ones (such as neutron).
From the quantum electrodynamics point of view, the spin is a distinction between the cariers of interaction (bosons, which have integer spin, such as photons) and the fermions that couple to the carriers of interaction via their charge and spin. It is not quite clear how one can throw away the spin in this picture without destroying it completely.
Finally, spin affects the interactions inexplicitly via the particle statistics, i.e. via the exclusion principle.
A: The charged pions, for example, $\pi^+$ and $\pi^-$, have zero spin but interact with a magnetic field, as can be seen from the curved tracks they leave in a bubble chamber with a magnetic field. So, to answer your question, spin is not theoretically necessary for electromagnetism. You could have a perfectly good and complete EM theory with spinless particles. But the real world doesn't work like that.
A: Macroscopically, in classical mechanics spin is the rotation of a body around an axis passing through it, and can be seen as  the angular momentum that the body has. Conservation of angular momentum is intrinsic in the classical theory .
When measurements started to look at the microcosm of atoms and particles, the conservations rules of energy, momentum were found to hold in the interactions, BUT unless an intrinsic angular momentum was  given to particles, conservation of angular momentum would not hold. So by experimental observations a fixed angular momentum, called spin, was assigned to all elementary particles so that the observations would fit the quantum theory that was developing. This has been validated over and over again with all measurements.
You ask:

how big of an role does spin play in the interactions which happen through the electromagnetic field?

The big role of keeping conservation of angular momentum in interactions .

Are there some interactions which spin-less particles cannot have, but those with spin can?

The fundamental interactions of elementary particles are four. Spin limits the possible interactions because of conservation of angular momentum, so there are differences between spin-non-zero-particle interactions  and spin-zero-particles, as with all conserved quantities, like charge, baryon number etc.
A: Magnetism is often perceived as the spin of subatomic particles that "makes" the magnetic dipole of these particles. Sometimes this complicates the understanding of magnetism on a macroscopic level as well as in the intraatomic interactions.
Classically (means, not in the QED‘s view) one may treat electrons and protons as having the two intrinsic properties charge and magnetic moment (electric field and magnetic field of the electron and the proton), that exist in the same way. Indeed, both properties are separable on the macroscopic level. The electric field by separating electrons from the nucleus and magnetic field by aligning atoms by their magnetic dipole moments. To put it directly: electrons and protons are just as much charges as magnets.

I know that spin is needed for defining the magnetic moment of any particle, and I have also read that the spin actually is the reason why some materials are magnetic.

Considering the above, the spin is - without considering the QED - a synonym for the magnetic moment. The aligned magnetic dipols of the involved subatomic particles are the reason for the macroscopic magnetic field.

What I want to know is whether spin is necessary for the some interactions in the electromagnetic field... in classical electromagnetic field theory, the electric and the magnetic fields could be considered as some combinations of partial derivatives of the vector potential (A). Any particle couples with the field and interacts with other particles through it.

There is a difference between the mathematics of an electro-magnetic-field dealt with in physics and the interactions resulting from these two phenomena. Particles as well as macroscopic bodies interact with their magnetic fields and with their electric fields. For example, an electron and a proton are attracted by their electric fields and aligned by their magnetic fields, but never the electric field interacts with the magnetic field.
Physics knows nothing about the internal structure of electric and magnetic fields. A current consists of electrons, water consists of H2O molecules, but we are not interested in the internal structure of E and B fields. The only thing we do is to represent them by field lines. And without caring about the components, we have introduced the model of virtual photons for the interactions. This makes things more difficult than they are. It is better to replace virtual photons by the interaction of charges and magnetic dipoles.
The QED was developed for processes on the atomistic level. The interaction between the nucleus and the electrons and between the electrons in the shells are both: electrical and magnetic interactions. But also here the term "spin" is interchangeable with the term „magnetic dipoles“. Somehow it is easier to imagine the paired electrons from the Paulis principle through paired magnets than through antiparallel spin.

Moving on, if we consider the quantum field theory version, we have two particles coupled with the electromagnetic field, which then interact with the exchange of bosons (photons). My question is: how big of a role does spin play in the interactions which happen through the electromagnetic field? Are there some interactions which spinless particles cannot have, but those with spin can?

Not having a spin for a particle is the same as not having a magnetic dipole. But perhaps the inner structure of spinless particles get influenced by an external magnetic field and induces a magnetic dipole. Only in this case the external field will deflect these moving particles (the Lorentz force) from the trajectory like electrons do.
A: The electromagnetism is pretty much wide field of knowledge and a lot of it goes well without the notion of any "spin".
The spin concept is usually first invoked when one needs to explain ferromagnetism (the ability of some substances to retain a static magnetic field after the external magnetic field is removed).
Unlike many ubiquitous electromagnetic phenomena that can be explained more or less classically, ferromagnetism escapes all classical attempts. That's why it is considered "purely quantum effect". The electrons' spin property is intimately involved in this effect.
A: EM has several levels of depth: Maxwell's eq. (about div and curl, infinitesimal conformal transformations of a field), which does not "see" the quantum phase, or monodromy of the EM connection; the later involves the vector potential yielding the EM field F=dA, and is essential, e.g. Aharonov-Bohm effect.
The spin of a particle signals the presence of a magnetic field, whatever the origin (e current model or A-flow) and interacts with a macroscopical EM-field, e.g. Stern-Gerlach experiment, Einstein de Haas effect and the reverse, Barnett effect, which both couple rotation and spin (magnetic moments).
These are "classical" effects involving the quantum structure (discrete) of matter, without the need of QED, because they "couple" macro and micro structure.
QED is needed for micro-micro interactions: electron-photon processes ("radiation"), scattering  etc. and spin is involved (e.g. atomic orbitals).
