Question 1: Let $G$ be a connected reductive group defined over a number field $K$, and $\pi$ is an irreducible cuspidal automorphic representation of $G(\mathbb{A}_K)$. Then by a theorem of Flath, we have $\pi=\otimes^{'}\pi_v$. All but finitely many $\pi_v$ are unramified and they all generic by Shalike.

I want to know whether there exists a finite place $\mu$ such that $\pi_{\mu}$ is supercuspidal for $G(K_{\mu})$ or there is no relation with cuspidal automorphic representation for $G(\mathbb{A}_K)$ and supercuspidal representation of $G(K_{\mu})$.

Question 2: Let $S$ be a finite set containing infinite places, and assume $\pi_v$ is unramified for $v \notin S$ and $\pi_{\mu}$ is generic for $\mu \in S$, then can I say $\otimes_{v}^{'}\pi_v$ is a cuspidal automorphic representation of $G(\mathbb{A}_K)$? If not, what additional condition should I add?

Question 3: I know there is a proposition below proving by Bernstein-Zelevinsky classification.

$\pi_v$ is an unramified and supercuspidal for $GL_n(K_v)$, where $v$ is a finite place if and only if $n=1$ and $\pi_v$ is an unramified quasi-character.

I want to know whether the proposition holds for general connected reductive p-adic group.

anyfinite place. $\endgroup$ – David Loeffler Mar 2 '18 at 14:29