How can you verify that a 3 by 3 unimodular matrix generates an infinite number of Fermat near misses?

Note : this answer has been simplified & improved at Jan 12, 9:16.

Here is an answer to your first question (the second one is much harder AFAIK).

Let

$$ M={\begin{bmatrix} 156625 & 115992 & −79656 \\\ 189000 & 139969 & −96120 \\\ 219624 & 162648 & −111695 \end{bmatrix}}, $$

$$ V_0= \begin{bmatrix} 3753 \\\ 4528 \\\ 5262 \end{bmatrix}, V= \begin{bmatrix} x \\\ y \\\ z \end{bmatrix}, d\begin{pmatrix}x \\\ y \\\ z \end{pmatrix} = x^3+y^3-z^3-1 $$

Then, one has the identity :

$$ d(M^3V)=184899d(M^2V)-184899d(MV)+d(V) \tag{1} $$ (you can check (1) by hand or with the help of a computer if you feel lazy).

It follows from (1) that $d(M^3V)=0$ if $d(V)=d(MV)=d(M^2V)=0$. By induction, we deduce $d(M^nV)=0$ (for $n\geq 0$) whenever $d(V)=d(MV)=d(M^2V)=0$. But it is straightforward to see that this is the case for $V=V_0$, and the result follows.

Note that the coefficients appearing in (1) are exactly the coefficients of the characteristic polynomial of $M$ ; note also that this polynomial is a "reciprocal" polynomial.