Are the Maxwell equations a correct description of the wave character of photons? In basic quantum mechanics courses, one describes the evolution of quantum mechanics chronologically. Interference experiments with particles showed that particles should have a wave character; on the other hand, the photo-electric effect showed that electromagnetic waves should have a particle character. So there's a particle-wave duality.
The course continues with postulating the Schrödinger equation, and solves the problems for the wave character of classical particles (e.g. with respect to harmonic oscillators to solve the UV-catastrophe in the description of blackbody radiation).
On the other hand, one describes the photo-electric effect extensively with perturbation theory, as the electromagnetic wave acts as a periodical force on an electron. But the electromagnetic wave is still described with the Maxwell equations, and not by Schrödinger.
Suppose history went a little differently: the corpuscular character would still be the generally accepted behaviour of light (as Newton preferred), and one discovered the wave character of photons and electrons simultaneously by doing interference experiments. The laws of Maxwell, Gauss, Faraday and Ampère weren't discovered yet. Schrödinger comes up with his equation, and one describes the wave character of electrons as well as photons with this equation. 
Would this equation give a exhaustive description of the electromagnetic (and vector like) behaviour of photons? In other words: is the Schrödinger theory, in some way, equivalent to the Maxwell theory for the description of photons? Or are the Maxwell equations some kind of limit for greater dimensions (like the Newton equations for mechanics)? What is the link between these two "wave character" descriptions of photons?
 A: 
Would this equation give a exhaustive description of the
  electromagnetic (and vector like) behaviour of photons? In other
  words: is the Schrödinger theory, in some way, equivalent to the
  Maxwell theory for the description of photons? Or are the Maxwell
  equations some kind of limit for greater dimensions (like the Newton
  equations for mechanics)? What is the link between these two "wave
  character" descriptions of photons?

A photon is an excitation of a "mode", i.e. a solution of Maxwell's equations satisfying the appropriate boundary conditions.  For example, a field constrained to be within a cavity has to satisfy the boundary conditions determined by the cavity.  A field in free space might be a spherically symmetric solution, depending on the source properties etc.  Once you have chosen your solution you can, in principle (in practice this may be very tricky - unless you're doing particle physics rather than quantum optics!), put a single excitation into it, creating a one-photon state.
Now, although the mode is a solution of Maxwell's equations, the state (at least in the Schroedinger picture), satisfies a Schroedinger equation.  This is just equivalent to saying that it evolves unitarily in time.
This Schroedinger equation, however, isn't the "wave equation for the photon" in the same way that in single-particle quantum mechanics the Schroedinger equation is the wave equation for the particle.  Rather, it's the time evolution equation for the state, which takes place in Hilbert space, not in spacetime.  Trying to emulate the single particle QM description by constructing a wavefunction for the photon is difficult:
The wavefunction would be the inner product of the state with position eigenstates $|x\rangle$ $$ \Psi(x)=\langle x|\Psi\rangle$$ The difficulty comes about because there isn't a(n undisputed!) Lorentz invariant position operator $\bf{\hat{x}}$ for photons.  
However, we can create single photon states.  These, however, are not really "like" the classical field which corresponds to the mode you excited.  See Lubos' answer here for discussion of the electric field in a single photon state for example.  If you want something which looks like the classical field, you need to construct the corresponding coherent state.  This does not have a definite number of photons.
A: Following the comment by Alfred Centauri, let me suppose that you discussed the Dirac theory of electron, and not the Schrödinger one. I will come back later on the possibility to describe matter-light interaction using the Schrödinger equation.
The Dirac theory describes the (special) relativistic behaviour of a particle. When complemented by the principle of gauge invariance (in particular the substitution of the normal derivative by the covariant one), it gives the basic playground for the simultaneous descriptions of the electromagnetic field (Faraday law and absence of magnetic monopole), the charge associated to the relativistic particle (the equation which replaces the Newton equation with the Lorentz force if you wish, but this has to be though with care) and their coupling (equations similar to the Maxwell-Ampère and Gauß, but there the current and charge densities have full quantum meaning, no more fluid interpretation as for the classical electromagnetism).
Obviously everything get more complicated when you try to quantise the electromagnetic field. The previous discussion didn't discussed the appearance of the photon.
I would say that the Wikipedia page related to the Dirac equation is not so helpful for understanding this point, but you could try to open the book by A. Messiah Quantum mechanics (volume II if not in an edition with the two volumes in one book), which contains all the pedagogical details you need, including the quantisation of the electromagnetic field in term of photon.
Schrödinger vs. Dirac description of matter
One can also describe the interaction between matter and light using the Schrödinger equation for the atom. This is the main study of the book

C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg Photons and Atoms: Introduction to Quantum Electrodynamics, Wiley (1992)

that I suggest you to read. In short, when the magnetic-like interaction is weak, the description using the Schrödinger equation is sufficient. You can understand this with the pictorial idea: when the (continuous laser) light field does not interact too much with the (gas) atoms, the effect can be describe by the first order term in interaction, which is already given by the Schrödinger prescription.
Historical perpectives
Now, regarding your historical perspective, it seems highly not probable that Dirac would have described the coupling between electrons and photons if he were unaware of the Maxwell's equations. This is once again because the gauge invariance is crucial in deriving the coupling. You may find more details about the history of gauge theory in the excellent collection of historical articles by

L. O'Raifertaigh The dawning of gauge theory, Princeton series in Physics (1997).

The same reasoning apply to the Schrödinger equation, because all these physicists were deeply influenced by the notion of field, that Maxwell really invented half a century before.
In short, the gauge invariance is the main ingredient of matter-field interaction, not the equation you're using to include it.
To be also noted:

*

*The particle behaviour of light was not the generally accepted behaviour of light (as you said) at the time of Maxwell's equations. Indeed, the Young two-slits experiment was already known by the end of the 18-th century.


*You do not really need to quantise the photon field to understand the photoelectric effect. This is discussed in a paper

Lamb, W. E., & Scully, M. O. The photoelectric effect without photon, in Polarisation, matière et rayonnement (pp. 363–369). Presses Universitaires de France (1969).

where they calculate the photoelectric effect quantising only the electron / detector behaviour.
A: 
Are the Maxwell equations a correct description of the wave character of photons?

Yes, Maxwell's equations are the wave equation for a photon, just as the Schrodinger equation is the wave equation for a nonrelativistic electron.

In other words: is the Schrödinger theory, in some way, equivalent to the Maxwell theory for the description of photons?

As others have pointed out, you can't apply the Schrodinger equation $i\hbar d\Psi/dt=-(\hbar^2/2m)\nabla^2\Psi$ to photons, because the Schrodinger equation is nonrelativistic, and a photon is never nonrelativistic.

Suppose history went a little differently: the corpuscular character would still be the generally accepted behaviour of light (as Newton preferred), and one discovered the wave character of photons and electrons simultaneously by doing interference experiments.

This is a very good question. There is a very nice discussion of this sort of thing in Peierls, Surprises in Theoretical Physics, section 1.3:

One of the most basic ideas of quantum mehcniacs is the analogy between light and matter [...] From this, it might appear an accident of history that physicists originally encountered only the wave aspects of light and only the corpuscular aspects of particles with mass, such as electrons. It therefore comes as a surprise to discover that this is no accident at all, and that the analogy between light and matter has very severe limitations.

I won't try to provide an analysis as complete as Peierls' in a physics.SE answer, but a crucial point is  that because photons are bosons, you can have a coherent superposition of photons in which a large number of photons are packed into a volume equal to a cubic wavelength. Such a superposition can have a well-defined amplitude and phase that can be measured by classical measuring devices such as antennas.
But you can't do this with electrons, because they're fermions. This is why the electron wavefunction isn't a classical field that can be measured directly.
A: The photon is an elementary particle.
Elementary particles are described by their rest mass, spin and quantum numbers as seen in the table. They then have attributes that are variable, like energy and momentum,  which are states measured with the corresponding quantum mechanical operator.
The photon has spin 1, zero mass and zero charge. Its energy is given by 
E=h*nu, where nu is the frequency characterizing it .
The frequency  is the one common point with the  classical electromagnetic light description. This is verified with the double slit experiment which displays the probabilistic nature from quantum mechanics when the photons are projected one by one and the interference pattern builds up . It is exactly the interference pattern given by the solutions of the classical Maxwell equations of electromagnetic waves.
Somehow, the large ensemble of photons contained in the light wave are coherently producing the electromagnetic wave studied macroscopically. Of course this is necessary for a self consistent physical theory. The macroscopic mathematical forms  have to emerge from the microscopic theories.
So yes, Maxwell's equations do describe photons since they give a light wave with frequency nu, the defining property of each photon in the ensemble. 
@LubošMotl has an article on his blog that shows how from the second quantization description of photons the classical electromagnetic wave emerges, similar to how thermodynamic quantities emerge from statistical mechanics.
The Schrodinger equation in this sense is not equivalent to the Maxwell's equations.
