I'm trying to learn the basics of the representation theory of $p$-adic groups and I'm stuck on a few things:

Let $G$ is a connected split reductive group over a non-archimedean local field $F$, and $K=G(\mathcal{O})$, a hyperspecial maximal compact subgroup of $G(F)$. Then there are two facts that apparently go hand in hand:

1. The Iwahori-Hecke algebra $H(G//K)$ is commutative.

2. If $\pi$ is any irreducible admissible (complex) representation of $G(F)$, then dim$(\pi^K) \leq 1$.

For the first statement there is a trick: one uses the Cartan decomposition to show that there is an anti-automorphism on the Hecke algebra that is the identity. But I don't see how to prove the second fact -- does it follow quickly from the first fact?

There is also this statement: For any irreducible admissible $\pi$, $\pi^K \neq 0$ if and only if $\pi$ is a Jordan-Holder component of an unramified principal series representation of $T(F)$, where $T$ is a maximal split torus in $G$.

I would appreciate any help on how these statements are proven. The source I'm using is a video lecture by Dipendra Prasad: the link is https://www.youtube.com/watch?v=4nDmAWix1kI.