Question: I am looking for an example of two random variables $X,Y$ such that 

(a) $X,Y$ are not independent.

(b) At least one of $X,Y$ is not normal.

(c) $E(X|y)$ (expected value of $X$ given $Y=y$) is linear in $y$, i.e. of the form $a+by$, and $E(Y|x)$ is linear in $x$.

(d) The correlation coefficient $\rho\neq \pm 1$.

Answer: 
Let $Y$ be any discrete random variable, choose constants $a$ and $b$, and just set $X=aY+b$. They are not independent, not Gaussian, and the conditional expectations are linear as you wanted.
Note that if Y is a binary random variable, then *any* function of Y is linear in Y. For example 
$$
\sin(Y)\;=\;\sin(0)\;+\;(\sin(1)-\sin(0))\cdot Y.
$$
This means that any generic distribution over $\{0,1\}^2$ satisfies your conditions. 

By the way, in general, if $E(X|Y)$ is linear in $Y$, then it must be necessarily of the form
$$
E(X|Y)\;=\;E(X)\;+\;\frac{{\rm cov}(X,Y)}{{\rm var}(Y)}(Y-E(Y)).
$$