  Fractional quantum Hall (FQH) system at Landau level filling fraction
$\nu=5/2$ has long been suggested to be non-Abelian, either Pfaffian (Pf) or
antiPfaffian (APf) states by numerical studies, both with quantized Hall
conductance $\sigma_{xy}=5e^2/2h$. Thermal Hall conductances of the Pf and APf
states are quantized at $\kappa_{xy}=7/2$ and $\kappa_{xy}=3/2$ respectively in
a proper unit. However, a recent experiment shows the thermal Hall conductance
of $\nu=5/2$ FQH state is $\kappa_{xy}=5/2$. It has been speculated that the
system contains random Pf and APf domains driven by disorders, and the neutral
chiral Majorana modes on the domain walls may undergo a percolation transition
to a $\kappa_{xy}=5/2$ phase. In this work, we do perturbative and
non-perturbative analyses on the domain walls between Pf and APf. We show the
domain wall theory possesses an emergent SO(4) symmetry at energy scales below
a threshold $\Lambda_1$, which is lowered to an emergent U(1)$\times$U(1)
symmetry at energy scales between $\Lambda_1$ and a higher value $\Lambda_2$,
and is finally lowered to the composite fermion parity symmetry
$\mathbb{Z}_2^F$ above $\Lambda_2$. Based on the emergent symmetries, we
propose a phase diagram of the disordered $\nu=5/2$ FQH system, and show that a
$\kappa_{xy}=5/2$ phase arises at disorder energy scales $\Lambda>\Lambda_1$.
Furthermore, we show the gapped double-semion sector of $N_D$ compact domain
walls contributes non-local topological degeneracy $2^{N_D-1}$, causing a
low-temperature peak in the heat capacity. We implement a non-perturbative
method to bootstrap generic topological 1+1D domain walls (2-surface defects)
applicable to any 2+1D non-Abelian topological order. We identify potentially
relevant spin TQFTs for various $\nu = 5/2$ FQH states in terms of fermionic
version of U(1)$_{\pm 8}$ Chern-Simons theory $\times \mathbb{Z}_8$-class
TQFTs.
