Teichmuller space for surface with cone points

The canonical reference, from which all follows, is Marc Troyanov's beautifully written paper:

Troyanov, Marc, Les surfaces euclidiennes à singularités coniques. (Euclidean surfaces with cone singularities), Enseign. Math., II. Sér. 32, 79-94 (1986). ZBL0611.53035.

If you are interested in hyperbolic surfaces with cone points, a very nice reference is Chris Judge's paper and references therein (though for the basics, there is not much difference between the Euclidean and hyperbolic cases).

Judge, Christopher M., Tracking eigenvalues to the frontier of moduli space. I: Convergence and spectral accumulation, J. Funct. Anal. 184, No. 2, 273-290 (2001). ZBL1005.58012.

The first theorem you want (that the Teichmüller space exists) is given as a remark in "Three-dimensional Orbifolds and Cone-Manifolds" by Daryl Cooper, Craig Hodgson, and Steve Kerckhoff. See Chapter 4, Section 6, second paragraph. They give further references, that may have more precise statements.

It seems to me that this covers surfaces with geodesic boundary, as well (and even cone points at the boundary - doubling reduces everything to the closed case). Thus the second theorem you want (FN coordinates) follows.

Here are some recent papers:

Rafe Mazzeo, Hartmut Weiss arXiv:1509.07608 Teichmüller theory for conic surfaces

arXiv:1710.09781 Rafe Mazzeo and Xuwen Zhu, Conical metrics on Riemann surfaces, I: the compactified configuration space and regularity.