The answer to these questions is not simple in general. Let me give an example related to your third question that shows some kind of situation we can encounter.

Let $C$ be a smooth curve of genus $3$ and let $S:= \mathrm{Sym}^2(C)$ be its second symmetric product. Then $S$ is a smooth, minimal surface of general type with $p_g=q=3$, $K^2=6$.

Therefore $\mathrm{Alb}(S)$ is an abelian threefold, and we can show that the Albanese map $$a_S \colon S \longrightarrow \mathrm{Alb}(S)$$ is a birational morphism onto its image $\Sigma \subset \mathrm{Alb}(S)$, which is a principal polarization.

Moreover:

 - if $C$ is non-hyperelliptic then $a_S$ is an immersion and so $\Sigma$ is smooth; 
 - if $C$ is hyperelliptic then $a_S$ contracts the unique
   $(-2)$-curve in $S$ corresponding to the $g_2^1$ on $C$; in this
   case, $\Sigma$ has an ordinary double point as its unique
   singularity.