Balanced Growth Path (Qualifier Question)
(That one can use the log-difference approximation for the growth rates, can be glimpsed by the fact that while the model apparently is set in discrete time, the log-evolution of technology is expressed using the exponential, which is how we express a constant growth rate in continuous time).  

From
$$\frac{y_{t+1}}{y_t}=e^\mu\bigg(\frac{k_{t+1}}{k_t}\bigg)^\alpha\qquad(3)$$

Taking logs you get

$$g_y = \mu + \alpha g_k$$

Where the "$g$"s are growth rates. One the balanced growth path, they have to be equal to some constant, say $g^*$, so you have

$$g^* = \mu + \alpha g^* \implies g^* = \frac {\mu}{1-\alpha}$$

which is the growth rate of the economy on the balanced growth path.

You should also be able (by simmply manipulating the equation with which you start showing your work), to arrive easily at

$$\left (\frac {c}{y}\right)^* = 1- (\delta +g^*)\left (\frac {k}{y}\right)^*$$
So you need only to determine one of the two to obtain the other.

Realize how you can write the left-hand side of $(1)$ while on the balanced growth path, then turn the equation around and solve for $k/y$. Don't expect to get something necessarily "simple-looking".