I realize that it is relatively easy to compute the variance of an AWGN convoluted with a sine-wave through auto-correlation function.
My question is how do I find the pdf if I know the variance and mean?
How can I prove that after all the math, my random variable still remains Gaussian?
Any help would be appreciated!
A linear time invariant system applied to a Gaussian process (such as AWGN) will give a Gaussian output. You've found the covariance function and the mean is also easy to calculate - if $L$ is linear, then $E[L(x(t))] = L(E[x(t)])$ where $x$ is the input (AWGN) and $L$ is the system (in this case, convolution with a sinusoid). The proof can be found in any introductory random processes book for engineers, such as Papoulis's Probability, Random Variables and Stochastic Processes or Stark and Woods' Probability, Statistics, and Random Processes for Engineers. The basic idea is to approximate the convolution as a Riemann sum, then use linear transformations of Gaussian vectors are still Gaussian.
The output autocorrelation function for a linear time invariant system is $R_{xy} (t_1,t_2)= L_2 [R_{xx}(t_1,t_2)]$ and $R_{yy} (t_1,t_2) = L_1[ R_{xy}(t_1,t_2)]$ where the $L_i$ notation means that the operator is applied with respect to the $t_i$ variable.
Since you have the mean function and covariance function (which the variance at $t$ is just the covariance between time $t$ and itself), and you know the output is Gaussian, you have the pdf at time $t$.