Here are the answers to your questions.

1.  The numerical class of the line bundle $\widetilde{L}$ is ${\rm deg}(L) \cdot x.$  To see why, note that the pullback of this class to $C^{d}$ must be ${\rm deg}(L)(X_{1}+ \cdots +X_{d}),$ where the divisor class $X_{i}$ on $C^{d}$ represents pullback of a point from the $i-$th projection. 

2.  The canonical bundle on $C_{d}$ is not $\widetilde{K}_{C},$ but rather $\widetilde{K_{C}}(-{\Delta}/2),$ where $\Delta$ is the diagonal divisor on $C_{d}.$