Don't forget that the conditional mean should be written $\mu(y)=\int x f_{X|Y}(x|y)d x$ and is a function of $y$.
For the posterior mean,
\begin{eqnarray*} E \left[ \theta |x \right] & = & \int \theta f_{\Theta |X} \left( \theta |x \right) \mathrm{d} \theta\\ & = & \frac{1}{f_X \left( x \right)} \int \theta f_{\Theta} \left( \theta \right) f_{X| \Theta} \left( x \left| \theta \right. \right) \mathrm{d} \theta \end{eqnarray*}
If you have two components, then \begin{eqnarray*} E \left[ \theta |x \right] & = & \left(\begin{array}{c} E \left[ \theta_1 |x \right]\\ E \left[ \theta_2 |x \right] \end{array}\right)\\ & = & \left(\begin{array}{c} \int \theta_1 f_{\Theta |X} \left( \theta |x \right) \mathrm{d} \theta_1\\ \int \theta_2 f_{\Theta |X} \left( \theta |x \right) \mathrm{d} \theta_2 \end{array}\right) \end{eqnarray*}
There is nothing special about vectors here. It was implicit, in $\begin{array}{lll} E \left[ \theta |x \right] & = & \int \theta f_{\Theta |X} \left( \theta |x \right) \mathrm{d} \theta \end{array}$ that, if $\theta$ is a vector then both sides of the equation are a vector.
For the mean from the joint distribution (and non-rigorously) \begin{eqnarray*} E \left[ X \right] & = & \int xf_{X, Y} \left( x, y \right) \mathrm{d} x \mathrm{d} y\\ & = & \int xf_X \left( x \right) \underbrace{\left( \int f_{Y \left| X \right.} \left( y|x \right) \mathrm{d} y \right)}_{= 1} \mathrm{d} x\\ & = & \int xf_X \left( x \right) \mathrm{d} x \end{eqnarray*}
So if the object in which you are interested only depends on some marginal variables, you only need to compute the mean with respect to the distribution of those marginal variables.
I hope that things are clearer now!