How to write BdG equation for generalize Hamiltonian?

I am reading this parer Majorana Corner Modes in a High-Temperature Platform,The Bogoliubov–de
Gennes Hamiltonian is :

$hat{H}=sum_{k}Psi^dagger_kH(vec{k})Psi_k$with

$Psi_k=(c_{a,kuparrow},c_{b,kuparrow},c_{a,kdownarrow},c_{b,kdownarrow},c_{a,-kuparrow}^dagger,c_{b,-kuparrow}^dagger,c_{a,-kdownarrow}^dagger,c_{b,-kdownarrow}^dagger)$
and Hamiltonian is $H(vec{k})=M(vec{k})sigma_ztau_z+A_xsin(k_x)sigma_xs_z+A_ysin(k_y)sigma_ytau_z+Delta(vec{k})s_ytau_y-mutau_z$.

All of the $sigma,tau,s$ are Pauli Matrix,$s$ is in spin space,$tau$ is about particle-hole symmetry and $sigma$ represent the orbital.

1. Question1:

I want to solve this model through BdG;I have known use Fourier Transformation make this Hamiltonian in real space.But I cannot confirm the correct process about this,I want to get the BdG form for this Hamiltonian,both k-space and real space are made me happy.

I download the Supplemental Material for this paper which have the Hamiltonian in real space,however I wish I can realize the detail about how to do this.

2. Quextion2:

In article has two image
I am a tiro in this,would someone willing to tell me how to plot these image?

Thanks for everyone who answer this question.