As John Rennie, says, what-does-a-de-broglie-wave-look-like has helpful answers which you should read first, but I don't think they are complete.
Do they behave like transverse waves?
No - the wave function for a single particle with no spin from the Schrodinger equation is just a scalar so there is no direction connected with it.
For example: can you polarize an electron beam?
You can polarise an electron (it has spin 1/2, so two options for spin direction). However, the spin part of an electron's wavefunction is separate from the spatial (wave) part - this is why the Schrodinger equation works for electrons even though it ignores spin. Thus the de Broglie wave itself is unaffected. (I believe this a valid alternative to the answer that a spin 1/2 particle has two de Broglie waves.)
Can you (internally) reflect and refract a particle beam? For example can you make a lens or prism the refract electron beams?
As yuggib mentioned, electron microscopes work by refracting electron beams. However, they use electric and magnetic fields in vacuum - so not much like ordinary refraction. Moreover, the lenses are well-described by classical physics. The problem is that the particles we are familiar with (other than the photon) have very short ranges in ordinary matter.
You can certainly diffract particle beams using crystals, in a similar way to the diffraction of x-rays by crystals, or light by a grating. The question is about refraction, though. See below.
can you tune a radio to receive very slow neutrons or electrons ..?
Please read this answer first. It turns out there are many fundamental differences.
- While a single particle has a wave function which can be a simple wave, the wave function of two particles is a function of the the positions of both particles (ie a function of 6 variables), etc. Not so easy to visualise.
- The wave function is a complex number. In fact, for a simple de Broglie plane wave, the modulus is constant (the particle can be anywhere) - only the argument varies (known as the phase in quantum mechanics).
- The argument (phase) can be changed without any physical change. For example, this answer by Lubos mentions that you can include or not include the rest energy of a particle ($E=mc^2$) in the formula $E=h\nu$, which changes the frequency, without changing the behaviour of the wavefunction. Clearly this wouldn't work with classical electromagnetic radiation - the tuning dial on the radio shows the frequency.
- The velocity (phase velocity) of a de Broglie wave is $c^2/v$ - so it equals c for a photon whose velocity $v$ is $c$, the speed of light, but is greater than $c$ otherwise. The group velocity (speed at which a wavepacket travels) is the speed of the particle. This isn't a problem, but shows how significantly massless particles like photons differ.
- Ron Maiman explains that
If you have many Bosons in a superposition state where they all share the same quantum state, their wavefunction becomes a classical field which obeys the Schrodinger equation.
He doesn't mention it, but I believe this starts to explain why light exists. Photons are bosons - particles that can be in the same quantum state, unlike electrons (fermions) for which the exclusion principle forbids this. In an ordinary beam of light there are many photons in the same or similar states. Somehow their collective wavefunction is manifested as real (not complex) electric and magnetic fields.
I therefore conclude the answer is no - you cannot use fermions such as electrons to produce a signal in an "electron radio".
I would be interested to know if @RonMaimon or @LubošMotl agree.