Some question about cupidal automorphic representation and supercuspidal representation 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.
 A: It is not necessary that a cuspidal $\pi$ have a supercuspidal local component: for example, let $\Delta(\tau)$ be Ramanujan's delta function and let $\pi = \otimes' \pi_p$ denote the associated cupsidal automorphic representation of $\textrm{GL}_2(\mathbb{A}_{\mathbb{Q}})$ (see Kudla's article here).  Since $\Delta$ has level 1, $\pi_p$ will be unramified for every prime $p$.  
What are your assumptions for question 2?  Given a random collection $\{\pi_v\}_v$ of irreducible local representations, almost all of which are unramified, the representation $\otimes' \pi_v$ will be neither cupsidal nor automorphic. 
There are many ways to prove question 3. For simplicity, assume $G$ is unramified over $K_v$ (that is, quasi-split and split over an unramified extension), let $K$ denote a hyperspecial maximal compact subgroup, $I\subset K$ an Iwahori subgroup, and let $P = MN$ be a minimal parabolic subgroup.  If $\pi$ were both ($K$-)unramified and supercuspidal, then Theorem 3.7 in Cartier's article in the Corvallis proceedings would give
$$0\neq \pi^K \subset \pi^I \stackrel{\sim}{\rightarrow}(\pi_N)^{K\cap M}.$$
If $N\neq 1$, then the right-hand side would be zero by supercuspidality.  Therefore $G$ must be a torus.
