Reference on spectral fractional Laplacian Are  there Harnack type inequalities and Schauder type estimates for the spectral fractional Laplacian. References are welcome. 
 A: This seems like a reliable entry point to the literature: What Is the Fractional Laplacian?

The purpose of this work is two-fold: (i) to give a comprehensive
  report of the commonly used definitions of the fractional Laplacian
  and examine their differences in bounded domains, and (ii) to
  quantitatively assess the available numerical methods developed for
  each definition. Of significance is the inclusion of recent work on
  implementing methods for nonzero boundary value problems. This work
  may be of use to practitioners looking to gain insight into which
  fractional Laplacian definition and associated numerical methods may
  be appropriate for their application. We present new solvers and
  discuss our implementation of existing numerical methods for
  discretizing the fractional Laplacians.

A: A quick Google Scholar search returns "An extension problem related to the fractional Laplacian" by L. Caffarelli and L. Silvestre, an online preprint of which can be found on Arxiv,
which itself cites "Properties of the solutions of the linearized Monge-Ampère equation  by L. Caffarelli and C. Gutiérrez".
However it's not clear to me that the latter actually proves an Harnack inequality for fractional operators.
A: On the probability side, this operator was extensively studied by Zhen-Qing Chen, Panki Kim, Renming Song and Zoran Vondraček. I always thought that Harnack inequality was proved in already in their first paper on that subject, Potential theory of subordinate killed Brownian motion in a domain (by R. Song and Z. Vondraček). Only now I realised it is not there.
It appears Harnack inequality was first proved by P. Kim and Ante Mimica in Harnack inequalities for subordinate Brownian motions. A general version, and much more, can be found in more recent Potential theory of subordinate killed Brownian motion.
All papers mentioned above can be found on arXiv, as well as on websites of the authors.

Regarding Schauder estimates, I do not think this is available in the probability literature. Quite likely the result can be found on the PDE side, which, unfortunately, I do not know well.
Edited: On the PDE side, Harnack inequality is proved in Harnack's inequality for fractional nonlocal equations by Pablo Raúl Stinga and Chao Zhang. Quite likely follow-ups to that paper contain regularity results you are interested in.
