The physics paper regularizes the volume and I don't expect a straight forward translation between the Selberg trace formula setting for finite volume Riemann surface and the regularized upper halfplane setting (not involving a Fuchsian group at all). The measure $\lambda \tanh(\lambda)$ is closely related to the Plancherel measure of $\mathbb{H}=SL_2(\mathbb{R})/SO(2)$, which naturally turns up in the spectral analysis of the Casimir operator of $\mathfrak{sl}_2$ from which you can derive the Laplacian.
So no, $\Gamma= \{1\}$ is not allowed in the context of the Selberg trace formula because $\Gamma \backslash \mathbb{H}$ needs to be finite-volume with respect to the measure $\frac{dxdy}{y^2}$.
For your remainig questions, I suggest additional references: Iwaniec- "Spectral theory of automorphic forms" Chapter 10 or Deitmar-Echterhoff - "Principles of Harmonic Analysis" Chapter 11 or Hejhal "The Selberg trace formula Volume I for $PSL(2, \mathbb{R})$".
6.7 is a definition. Usual $K$ denotes kernel.
6.8 is a partition into conjugacy classe. I think that $\langle g \rangle$ should denote the centralizer. $g = \gamma_0$ is more common, since the centralizer is a cyclic group. Conjugating by elements from the centralizer doesn't add anything new;)
For the last equation, look at the suggested reference.