Q1. Mathjax matrix not registering newlines when mixed with markdown text

A1. This is caused by rendering \\ as \ by default in markdown parser. And \ is an escape character commonly used across many programming languages.

You could use a \ before every actual \ you want to use. So if you wanted \begin{bmatrix} you would have to write \\begin{bmatrix} and if you want \\ you need to use \\\\.

$ \\begin{bmatrix}a & b \\\\ c & d\\end{bmatrix} $