What are the most efficient particle acceleration mechanism in space and astrophysical plasmas. Wave-particle interaction, shocks, magnetic reconnection?
It depends on what you mean by efficient. Are you asking which method can accelerate the most particles or which can accelerate particles to the highest energies? Or which method has the most efficient transfer of energy?
For the sake of argument, I am going to assume you are asking about which mechanism can generate the highest energy particles.
In principle, magnetized shock waves are thought capable of generating some of the highest energy particles in the universe through something called diffusive shock acceleration or DSA. There's also a good article by Sironi et al. [2013] that discusses the upper energy limits for particle energization by astrophysical shocks. However, Sironi et al. [2015] argues that the upper energy limit for shock acceleration tops out around 1017 eV (i.e., ~0.016 J).
Magnetic reconnection is slightly less clear because there are two primary mechanisms that can accelerate particles: the reconnection electric fields that drive the jets or the merging magnetic islands which act like scattering centers similar to DSA for shocks. The outflow jets are limited by the Alfven speed, which can approach the speed of light in the right limits. The magnetic island merging would be limited by island size. That is, the upper limit on energization by DSA is limited by time spent near the shock or moving magnetic field gradient. The size of coherent magnetic islands are likely not going to be extremely large in most systems, which will likely limit the upper bounds on the energy of an accelerated particle. Even so, some systems seem to be capable of generating leptons with energies in excess of 1012 eV [e.g., see Treumann and Baumjohann, 2015]. While other systems may be able to inject particles into further energization by way of DSA starting at energies of 1016 eV [e.g., see Sironi and Benoit, 2017]. That is, in the Crab Nebula simulations suggest that reconnection may pre-energize particles up to 1016 eV through a runaway effect.
Wave-particle interactions (WPI) are more complicated and the zoo of possible waves makes it impossible to fully address this question. However, I can provide some case examples. An electromagnetic fluctuation called a whistler wave can exist over a rather extreme range of frequencies and wavenumbers (i.e., an example case of where a name was given and the parameters defining the fluctuations were too broad). They can be high frequency, short wavelength oscillations up near the electron cyclotron frequency or low frequency, long wavelength oscillations down near (and below) the lower hybrid resonance frequency. In the latter case, they are typically found to propagate obliquely to the quasi-static magnetic field, which means the projection of the wave vector, $\mathbf{k}$, has a parallel and perpendicular component magnitude. Since the phase speed direction depends on $\mathbf{k}$, the magnitude of the parallel and perpendicular phase speeds are given by:
$$
\begin{align}
V_{ph, \parallel} & = \frac{ \omega }{ k_{\parallel} } = \frac{ \omega }{ k \ \cos{\theta_{kB}} } \tag{0a} \\
V_{ph, \perp} & = \frac{ \omega }{ k_{\perp} } = \frac{ \omega }{ k \ \sin{\theta_{kB}} } \tag{0b}
\end{align}
$$
where $\omega$ is the fluctuation angular frequency and $\theta_{kB}$ is the wave normal angle. In the limit as $\theta_{kB} \rightarrow 90^{\circ}$, we see that $V_{ph, \parallel} \rightarrow \infty$. This is a little unphysical so you can think of it as approaching the speed of light in vacuum. Therefore, if one constructs a wave field with a broad range of $\omega$ and $\theta_{kB}$, one could, in principle, accelerate particles to ultra-relativistic energies if the wave had sufficient potential energy density (i.e., large enough amplitude). An example simulation doing just this to explain particle acceleration in solar flares was carried out by Cairns and McMillan [2005]. Observational evidence of this phenomena at lower energies was reported by Wilson et al. [2012] and later inferred to be responsible for generating the observed relativistic electrons by Wilson et al. [2016].
Another interesting extrapolation was carried out by Wilson et al. [2011]. They used the nonlinear expression for the maximum kinetic energy change of an electron interacting with a whistler wave in the radiation belts to determine that the extremely large amplitude waves presented in their study could increase an electron's kinetic energy by ~70 MeV (this is ignoring radiation losses during acceleration, among other things). The terrestrial radiation belts are not extreme in the grand scheme of things, so the upper bound in say, the Jovian radiation belts is likely much higher.
In summary, the evidence so far leans toward collisionless shock waves generating the highest energies of these possible acceleration mechanisms. However, note that DSA generally involves self-consistently generated waves by the particles that are undergoing DSA, i.e., the particles are reflected and stream against the incident flow generating instabilities that radiate electromagnetic fluctuations which then act as scattering centers for DSA. That is, none of the mechanisms are likely to act in isolation, e.g., reconnection regions are riddled with large amplitude electromagnetic fluctuations, collisionless shock ramps always have lots of large amplitude electromagnetic fluctuations, etc.
Does it depend on the energy of the particle?
In a plasma, the answer to this question is almost always yes. The reasons are multi-fold but you can see from the Lorentz force that it is dependent upon the particle velocity. In the case of WPI, these are always energy and pitch-angle dependent. We know that DSA is more efficient for particles starting with higher energies because to gain energy in DSA, the particle needs to cross the shock lots of times before it "runs away." That is, higher energy particles can cross the shock lots of times easily before the shock moves an appreciable amount whereas lower energy particles may not cross it more than once. As for reconnection, because it involves electric and magnetic field gradients, it's ability to energize particles, will as well, be inherently energy and pitch-angle dependent.
Also, are there mechanisms best suited for thermalizing the particle distribution (ie current sheets) and others that mostly accelerate particles ie magnetic reconnection?
Note that a current sheet cannot energize or thermalize a particle velocity distribution. A current sheet is just a 2D flow of charged particles. The processes that dissipate that flow can and do energize particles but the current sheet itself does none of that.
As for what is best for thermalizing particles, that would be a combination of WPI and/or shock compression. Since collisionless shock waves always have large amplitude electromagnetic fluctuations present within and around them, it's extremely difficult to isolate the cause of the thermalization. Magnetic reconnection events tend to be, in the absence of WPI, much less efficient at thermalizing a particle distribution (e.g., see some of above references and references therein).
