Welcome to mathoverflow!

I believe I can answer question 3. My reference here is Walters, P. An introduction to Ergodic Theory. Chapter 6.2.

When $T: X \mapsto X$ is a continuous transformation of a compact metrisable space $X$, then there will always be a measure for which $T$ is measure preserving. Hence $M(T)$ is never empty. (Corollary 6.9.1)

The space $M(T)$ is compact, convex and nonempty. Hence it has an extreme point by this argument.

The extreme points are the ergodic measures (Walters theorem 6.10(iii))

Hence $E(T) \neq \emptyset$.

Welcome to mathoverflow!

I believe I can answer question 3. My reference here is Walters, P. An introduction to Ergodic Theory. Chapter 6.2.

When $T: X \mapsto X$ is a continuous transformation of a compact metrisable space $X$, then there will always be a measure for which $T$ is measure preserving. Hence $M(T)$ is never empty. (Corollary 6.9.1)

The space $M(T)$ is compact, convex and nonempty. Hence it has an extreme point by this argument.

The extreme points are the ergodic measures (Walters theorem 6.10(iii))

Hence $E(T) \neq \emptyset$.