Formula for weighted Pearson correlation can be easily found on the web, StackOverflow, and Wikipedia and is implemented in several R packages e.g. psych, or weights. It is calculated the same as regular correlation but with using weighted means,
$$ m_X = \frac{\sum_i w_i x_i}{\sum_i w_i}, ~~~~ m_Y = \frac{\sum_i w_i y_i}{\sum_i w_i} $$
$$ s_X = \frac{\sum_i w_i (x_i - m_X)^2}{ \sum_i w_i}, ~~~~ s_Y = \frac{\sum_i w_i (y_i - m_Y)^2}{ \sum_i w_i} $$
and weighted covariance
$$ s_{XY} = \frac{\sum_i w_i (x_i - m_X)(y_i - m_Y)}{ \sum_i w_i} $$
having all this you can easily compute weighted correlation
$$ \rho_{XY} = \frac{s_{XY}}{\sqrt{s_X s_Y}} $$
As about your second question, as I understand it, you would have data about correlations between political orientation and preference for the twenty artists and users binary answers about his/her preference and you want to get some kind of aggregate measure of it.
Let's start with averaging correlations. There are multiple methods for averaging probabilities, but there doesn't seem to be so many approaches to averaging correlations. One thing that could be done is to use z-transforms as described on MathOverflow, i.e.
$$ \bar\rho = \tanh^{-1} \left(\frac{\sum_{j=1}^K \tanh(\rho_j)}{K} \right) $$
it has more sophisticated rationale, but basically what taking tangents of correlation coefficients is it "flattens" the extreme values (see below) so they have lower influence on the final estimate (similar approach is often used for averaging probabilities).
Next, if you notice that if $r = \mathrm{cor}(X,Y)$, then $-r = \mathrm{cor}(-X,Y) = \mathrm{cor}(X,-Y)$, so positive correlation of musical preference with some political orientation is the same as negative correlation of musical dislike to such political orientation, and the other way around.
Now, let's define $r_j$ as correlation of musical preference of $j$-th artist to some political orientation, and $x_{ij}$ as $i$-th users preference for $j$-th artist, where $x_{ij} = 1$ for preference and $x_{ij} = -1$ for dislike. You can define your final estimate as
$$ \bar r_i = \tanh^{-1} \left(\frac{\sum_{j=1}^K \tanh(r_j x_{ij})}{K} \right) $$
i.e. compute average correlation that inverts the signs for correlations accordingly for preferred and disliked artists.