Nonvanishing of central L-values of quadratic twists? Let $\pi$ be a cuspidal automorphic representation of GL(2) over a number field (if you want, assume it's $\mathbb Q$ and $\pi$ comes from a holomorphic modular form).  
In the case $\pi$ has trivial central character, the epsilon factor determines the parity of order of vanishing of $L(1/2,\pi)$.  If $\chi$ is a quadratic character, and $\pi$ in fact comes from an elliptic curve, then one expects $L(s,\pi \otimes \chi)$ to have rank 0 half the time and rank 1 half the time (Goldfeld's conjecture).

(1) Is there a precise generalization of Goldfeld's conjecture to more general $\pi$ (assume what you need)?

I know there are several nonvanishing results and bounds on proportions for rank 0 and rank 1 if $\pi$ has trivial central character.  However if $\pi$ does not have trivial central character (and is not self-dual), then I know little more than that Friedberg-Hoffstein says $L(s,\pi \otimes \chi)$ has rank 0 infinitely often.

(2) Is anything else known/expected when $\pi$ is not self-dual?

I know nothing about Katz-Sarnak philosophies and Random Matrix Models, but do they apply for non-self-dual representations?
 A: Though it is perhaps not an "answer" as such, let me try to explain some intuition.
In certain settings, it is possible to formulate subtle analogues of Mazur's conjecture
for nonvanishing of central values (think Cornut-Vatsal) from the refined conjecture 
of Birch and Swinnerton-Dyer via Iwasawa theory. A general method is explained in 
section 4 of Coates-Fukaya-Kato-Sujatha,  " Root numbers, Selmer groups, and 
non-commutative Iwasawa theory" (available at http://www.math.tifr.res.in/~sujatha/root.pdf). 
CFKS consider the setting of the so-called False-Tate curve extension, but
 a similar (and simpler) set of arguments can be used to deduce an analogous
 conjecture for the setting of the ${\bf{Z}}_p^2$ of an
 imaginary quadratic field. To be slightly more precise, fix a rational prime $p$.
 Fix an eigenform $f \in S_2(\Gamma_0(N))$ with $(N,p)=1$. Fix an imaginary
 quadratic field $k$ with discriminant prime to $pN$. Let $k_{\infty}$ denote the 
 ${\bf{Z}}_p^2$-extension of $k$, which is the compositum of the cyclotomic
${\bf{Z}}_p$-extension $k^c$ with the anticyclotomic ${\bf{Z}}_p$-extension $k^a$.
Write $\lambda_f(k)$ to denote the cyclotomic $\lambda$-invariant associated to 
$f$, with $\mu_f(k)$ the cyclotomic $\mu$-invariant. Let $\mathcal{W}$ be any finite
order character of $\operatorname{Gal}(k_{\infty}/k)$. Such a character can always be written as 
a product of characters $\rho \cdot \chi$, where $\rho$ is a finite order character
of $\operatorname{Gal}(k^a/k)$, and $\psi$ is a finite order character of $\operatorname{Gal}(k^c/k)$. What the 
CFKS conjecture predicts, very roughly, is the following assertion. Assume that 
$\mu_f(k)=0$, and fix a finite order character $\rho$ of $\operatorname{Gal}(k^a/k)$. Let $\Psi$
denote the set of finite order character of $\operatorname{Gal}(k^c/k)$. Then, assuming the 
refined Birch and Swinnerton-Dyer conjecture, \begin{align*}
\sum_{\psi \in \Psi} \operatorname{ord}_{s =1}L(f \times \rho \cdot \psi, s) &\leq \lambda_f(k).
\end{align*} So, what does this tell us? Well for one, it tells us that Rohrlich 
nonvanishing (at least in this setting) should be a general phenomenon. 
One can make this intuition slightly more precise via the following heuristic
argument. View any finite extension of $k^c$ over $k$ as a totally imaginary
quadratic extension of its maximal totally real subfield. Suppose that the 
nonvanishing theorem of Cornut-Vatsal were uniformly effective (in the sense
that their $n$ sufficiently large could be replaced by some absolute $n_k$ that does not
grow as we ascend the cyclotomic tower). Then, invoking their result systematically
and decomposing via Artin formalism, we should (I think) expect the following behaviour.
Let $\epsilon(f/k, s)$ denote the root number of the Rankin-Selberg $L$-function 
$L(f/k, s)$. Given $\rho$ a finite order character of  $\operatorname{Gal}(k^a/k)$ of conductor greater that $p^{n_k}$, we should have: 
\begin{align*}
\sum_{\psi \in \Psi} \operatorname{ord}_{s=1} L(f \times \rho \cdot \psi, s) &= 0 \text{ if $\epsilon(f/k, 1) = 1;$}   \end{align*}
\begin{align*}
\sum_{\psi \in \Psi} \operatorname{ord}_{s=1} L(f \times \rho \cdot \psi, s) &= 1 \text{ if $\epsilon(f/k, 1) = -1;$}   \end{align*}
Sorry to have skipped steps, or if this is perhaps somewhat unclear in places. The main idea is that there are subtle generalizations of Mazur's conjecture to the types of settings that you will likely want to consider. These generalizations suggest that one should expect generic nonvanishing à la Rohrlich, even in the case where the root number at the bottom is $-1$. And though it is not a priori clear, it might be possible to use these sorts of ideas to obtain the general formulation of Goldfeld's conjecture that you ask for.
A: (1) In a "true" Katz-Sarnak context, i.e., over finite fields, non self-dual situations definitely make sense, exist, and are among those studied in their book (chapters 9 and following). In fact the basic question of proportion of vanishing  v.s. non-vanishing is really an application of Deligne's Equidistribution Theorem.  Basically, when the size of the field goes to infinity, if you have a known (compactifo-complexified) monodromy group, the proportion of vanishing at the central point will converge to the Haar probability -- in the monodromy group -- of matrices with $1$ as an eigenvalue. In most cases this is computable once the monodromy is known. However, this story has the usual limitation (field goes to infinity can not be replaced, for the moment, with size of matrices goes to infinity).
(2) For L-functions over $\mathbf{Q}$, the analytic methods which give a "decent" proportion of non-vanishing for quadratic twists (not positive proportion, but about $1/(\log D)^A$ for twists by quadratic characters of size up to $D$, for some fixed $A$) work whether the form is self-dual or not. But one definitely expects that twists of a fixed form which is not self-dual should have "density" zero of order of vanishing at least $1$ (this is as precise as Goldfeld's Conjecture).  In many cases, it might well be that no twist vanishes at $1/2$, or at most finitely many.
A: I suggest you try asking Dipendra Prasad. I have some vague memory that he once told me that (conjecturally) if pi is not self dual then the central value will never vanish.
