Question on stochastic matrix with stirctly positive entries

If you know a bit about Markov chains, your question is basically answered by Theorem 15.3 in https://www.math.ucdavis.edu/~gravner/MAT135B/materials/ch15.pdf. It asserts that something stronger is true, which I will restate in simpler terms here:

Theorem: Suppose $P$ is a $n\times n$ stochastic matrix with strictly positive entries $(P_{ij})_{1\leq i,j\leq n}$. Then, $$\lim_{n\to\infty} P^n=A,$$ where $A$ is a $n\times n$ matrix such that in each column of $A$, all entries in that column are equal.

In particular, for any $x\in\mathbb{R}^n$, the vector $Ax$ will have all of its entries equal to each other.

Yes and this is a consequence of the basic limit theorem of MC theory. This MC has a unique stationary distribution. $M^{n}$ converges to a matrix in which each row is the stationary distribution so $\lim M^{n} x$ has all its components equal (to $\sum \pi_i x_i$ where $(\pi_i)$ is the stationary distribution).