Let $C$ be a smooth projective curve over some field (algebraically closed if necessary), ${\rm Pic}^{d}(C)$ the Picard variety of degree $d$ divisor classes on $C$, $C^{(d)}$ the $d$th symmetric power of $C$. We want to define an Abel map $\alpha_{d}:C^{(d)} \rightarrow {\rm Pic}^{d}(C)$, which intuitively sends a point on $C^{(d)}$ thought of as an unordered $d$-tuple $x_1,...,x_d$ of points in $C$ to the class of the divisor $x_1+...+x_d$.

Questions:

Is $C^{(d)}$ a fine moduli space of effective degree $d$ divisors and ${\rm Pic}^{d}(C)$ a fine moduli space of degree $d$ divisor classes (or degree $d$ line bundles), so that there are appropriate universal families over them? Does the field of definition matter here?

I've heard that for $d \leq g$, the Abel map $\alpha_{d}$ is generically injective. Does this mean that it is injective on a Zariski open subset of $C$?

Consider the case $d=1$, so that the Abel map is $\alpha_1: C \rightarrow {\rm Pic}^{d}(C)$. Can one describe the image and fibres explicitly? What if $C$ is hyperelliptic? Then can we give such a description in terms of the map $C \rightarrow \mathbb{P}^{1}$?