The explanation is a bit long, but relies upon very standard material of the Teichmuller theory (much of it you probably know, but I am not sure). I will be addressing Question 1*, since Question 1 requires some clean-up, as formulation of the Teichmuller Theory in terms of qc maps $X\to X$ is artificial: One usually allows the target Riemann surface to vary.
Suppose that $G$ is a topological group. The relevant example for us is $G=PSL(2, {\mathbb R})$. To topologize it, one could take any reasonable topology, say the topology of pointwise convergence on the hyperbolic plane, or topology of uniform convergence on compacts in the hyperbolic plane, or topology coming from the matrix group topology of $SL(2, {\mathbb R})$. Let $\Gamma$ be a discrete group. One topologizes
$$
Hom(\Gamma, G)
$$
via topology of elementwise convergence of homomorphisms: This is the same as the subspace topology coming from the product topology on
$$
\prod_{\Gamma} G.
$$
If the group $\Gamma$ is finitely generated, with generating set $\{\gamma_1,...,\gamma_k\}$ then this topology is equivalent to the topology of convergence on $\{\gamma_1,...,\gamma_k\}$, i.e. the subspace topology coming from
$$
\prod_{i=1}^k G.
$$
We will use this fact later on.
Given a compact (connected) Riemann surface $X$ of genus $\ge 2$, I will abbreviate:
$$
Hom(X)= Hom(\pi_1(X), PSL(2, {\mathbb R})),
$$
Define
$$
R(X)=Hom(X)/PSL(2, {\mathbb R})
$$
where $PSL(2, {\mathbb R})$ acts on homomorphisms via conjugation: For $\alpha\in PSL(2, {\mathbb R})$ and $h\in Hom(\pi_1(X), PSL(2, {\mathbb R})$ define $\alpha\cdot h$ by
$$
\alpha \cdot h(\gamma)= \alpha h(\gamma) \alpha^{-1},
$$
where $\gamma\in \pi_1(X)$. (I am sloppy here regarding the basepoint in $X$.)
This quotient is actually quite terrible, but things improve quite a bit once we restrict to the subspace of discrete and faithful homomorphisms,
$$
Hom_{DF}(X)\subset Hom(\pi_1(X), PSL(2, {\mathbb R})),
$$
consisting of homomorphisms which have discrete image and are injective (faithful). Then $Hom_{DF}(X)$ is open in $Hom(X)$ and consists of two connected components, both preserved by the action of
$PSL(2, {\mathbb R})$. Moreover, this action is free and proper.
Set
$$
R_{DF}(X)= Hom_{DF}(X)/PSL(2, {\mathbb R}).
$$
This space consists of two components both of which are smooth manifolds of dimension $6g-6$, where $g$ is the genus of $X$. I will denote $R_{DF}^+(X)$ the component whose elements are equivalence classes of homomorphisms induced by equivariant orientation-preserving homeomorphisms of the hyperbolic plane:
$$
h(\gamma)= \beta h(\gamma) \beta^{-1}
$$
where $\beta$ is a homeomorphism as above.
One has a homeomorphism
$$
\sigma: {\mathcal T}(X)\to R_{DF}^+(X)
$$
defined as follows: Take $[\mu]\in {\mathcal T}(X)$, let $f_{\mu}: X\to X_\mu$ be the quasiconformal homeomorphism given by $\mu$. I will use the marking on $X_\mu$ given by the homotopy class of $f_\mu$. Then let $\Gamma_\mu< PSL(2, {\mathbb R})$ be a Fuchsian group uniformizing $X_\mu$ (it is defined uniquely up to conjugation by elements of $PSL(2, {\mathbb R})$); lift $f_\mu$ to the hyperbolic plane, to a quasiconformal homeomorphism $F_\mu: {\mathbb H}^2\to {\mathbb H}^2$.
Remark. You use the notation $f_{\tilde\mu}$ for $F_\mu$.
Let $\Gamma< PSL(2, {\mathbb R})$ be the Fuchsian group uniformizing $X$; I will identify it with $\pi_1(X)$.
Then $h_{\mu}: \Gamma\to \Gamma_\mu$ is an isomorphism defined by the familiar formula
$$
h_\mu(\gamma)= F_\mu \gamma F_{\mu}^{-1}.
$$
One verifies that the correspondence $[\mu]\to h_\mu$ depends only on $[\mu]$, up to conjugation by elements of $PSL(2, {\mathbb R})$. Hence, we obtain a map
$$
\sigma: [\mu]\to [h_\mu]\in R_{DF}(X).
$$
Continuity of $\sigma$ is clear (since qc maps depend continuously on the Beltrami differential, subject to normalization at three distinct points in the boundary circle of the hyperbolic plane; this normalization exactly corresponds to working with equivalence classes of homomorphisms). One then proves that $\sigma$ is a bijection, by constructing an equivariant qc homeomorphism of the hyperbolic plane inducing the given element $h\in Hom_{DF}(\pi_1(X), PSL(2, {\mathbb R}))$. It follows that $\sigma$ is a homeomorphism, since its domain and range are manifolds of the same dimension.
Remark. There are other ways to see that $\sigma$ is a homeomorphism (quasiconformal stability), but the above proof is the "classical" one, from the early 1960s.
With all these preliminaries out of the way, let's go back to your question. You are assuming that $F_{\mu_n}$ is a sequence of qc maps of the hyperbolic plane as above, which converges (say, uniformly on compacts) to the identity homeomorphism $id$. Let $h_n: \Gamma\to \Gamma_n$ be isomorphisms induced by $F_{\mu_n}$, where $\Gamma_n$ is a Fuchsian group uniformizing the Riemann surface $X_n$ (the one given by $[\mu_n]$).
As a compact, I will take a compact fundamental polygon $P$ of the group $\Gamma$. Now, apply the equivariance condition
$$
h_{\mu_n}(\gamma_i)= F_{\mu_n} \gamma_i F_{\mu_n}^{-1}=\gamma_{i,n}.
$$
applied to the generators $\gamma_i$ of $\Gamma$ (the side-pairing transformations of $P$). Convergence of the maps $F_{\mu_n}\to id$ (even pointwise convergence would suffice!) implies that
$$
\lim_{n\to \infty}\gamma_{i,n}= \gamma_i.
$$
Thus, $\lim_{n\to\infty} h_n=id: \Gamma\to \Gamma$ (recall the discussion of topology on $Hom(X)$). Now, apply continuity of $\sigma^{-1}$ to conclude that
$$
\lim_{n\to \infty} [\mu_n]=0,
$$
where $0$ is the Beltrami differential of the identity map.
