I think that (1) is false.

Imagine 2 manifolds defined as:

$$(x_1,x_2) \in (0,1) \times (0,1)$$

and

$$(x_1,x_2,x_3) \in (0,1) \times (0,1) \times (0,1)$$

and you choose $x$ randomly.

Define the distance between points $(x,y)$ as as:

1 if $x_1 \neq y_1$

$|| x - y ||_2$ if $x_1 = y_1$

They are different manifolds because 1 is locally $R$ and 1 is locally $R^2$, but almost always $M_n = 1 - I^{n \times n}$, so the random matrix (and therefore it's eigenvalues) are the same.

I think that 1) is false.
Imagine 2 manifolds defined as:

$$(x_1,x_2) \in (0,1) \times (0,1)$$

and

$$(x_1,x_2,x_3) \in (0,1) \times (0,1) \times (0,1)$$

and you choose $x$ randomly.

Define the distance between points $(x,y)$ as as: $$ d(x,y) = \begin{cases} 1 & \text{if }x_1 \neq y_1,\\ \| x - y \|_2 &\text{if }x_1 = y_1. \end{cases} $$ They are different manifolds because 1 is locally $\Bbb R$ and 1 is locally $\Bbb R^2$, but almost always $M_n = 1 - I^{n \times n}$, so the random matrix (and therefore it's eigenvalues) are the same.