generating correlated random variables of different distributions via gaussian copula If I want to generate two random variables, one is normally distributed N ~N(10, 25) and the other one, E, is exponentially distributed with mean 1. I was not given a particular correlation coefficient. 
First I get two independent standard normals A, B. 
Then I generate a correlated normal C, with unknown correlation coefficient p through a linear combination of A, B,  $C = pA + \sqrt{1-p^2}B$.
To get N, I simply transform A such as $N = 10 + 5A$.
Then I use the normal CDF to get the $unif(0,1)$ variable corresponding with $C$, $F(C) = U$ where $U~unif(0,1)$. From inverse transform of exponential cdf, 
I get that E = -ln(1-U). 
Is this a correct approach? I am also not sure if given the information in the first two lines I could have inferred some correlation coefficient p. 

It is the correct approach to generate the desired marginal distributions.
Taking $N = 10+ 5A$ and $E = -\ln(1 - F(B))$ we get independent random variables where $N$ has the normal distribution (mean $10$, variance $25$) and $E$ has the exponential distribution (mean $1$).
The Gaussian copula is applied by using $E = -\ln(1 - F(C))$ where $C = \rho A + \sqrt{1 - \rho^2}B$. For a given specified $\rho \neq 0$, dependence is introduced.  The random variables $N$ and $E$ will have non-zero correlation but it is affected by other moments. We will not have $corr(N,E) =\rho$ precisely, but, rather a close approximation
$$corr(N,E) = \frac{\mathcal{E}[ \,(N - \mathcal{E}(n))(E - \mathcal{E}(E)) \, ]}{\sqrt{var(N)} \sqrt{var(E)}} \approx \rho$$
