Obviously, this is a homework assignment, so I don't want it done for me; however, I am confused. Perhaps I am just confused by the problem or the wording, but I am totally stuck on what to do.

I believe the output signal should be a convolution where Z = X + Y, and Y is the gaussian(0, 2). After I solve the convolution and receive Z, I don't know what to do.

My guess is that you need to solve for the conditional cumulative distribution [itex] F(w,y) = P(X + Y \le w | Y = y) = P(X \le w - y | Y = y) [/itex]. Then differentiate with respect to w to get the density. (I think you interpreted "gaussian noise channel" correctly in the context of the problem, but I think it has an interpretation as a continuous stochastic process in other contexts.)

If g(X,Y) is joint density of the convolution X + Y, can you set Y = y and integrate symbolically with respect to X from minus infinity to w-y to obtain [itex] P(X \le w -y| Y = y) [/itex]? (I haven't worked the details out, so I don't know.)

If output (Z)= noise(Y) + signal (X) then the following is the working principle:
Find the distribution of Z.
Then from the joint distribution Z and X find the distribution X|Z.