The first natural observation is the following. Set as usual $q(X):=h^{1, \,0}(X)$, so that $q(X)= \dim \, \mathrm{Alb}(X)$. Then the Albanese map of $X$ cannot be surjective if $\dim \,X < q(X)$, and it cannot be injective if $\dim \,X > q(X)$.
Moreover, the Albanese map is never injective if $X$ contains some rational curve, because any map from $\mathbb{P}^1$ to an abelian variety is necessarily constant.
Let me now give an example related to your third question, showing that the answer can be quite subtle in general.
Let $C$ be a smooth curve of genus $3$ and let $X:= \mathrm{Sym}^2(C)$ be its second symmetric product. Then $X$ is a smooth, minimal surface of general type with $p_g=q=3$, $K^2=6$.
Therefore $\mathrm{Alb}(X)$ is an abelian threefold, and we can show that the Albanese map $$a_X \colon X \longrightarrow \mathrm{Alb}(X)$$ is a birational morphism onto its image $\Sigma \subset \mathrm{Alb}(X)$, which is a principal polarization.
Moreover:
- if $C$ is non-hyperelliptic then $a_X$ is an immersion and so $\Sigma$ is smooth;
- if $C$ is hyperelliptic then $a_X$ contracts the unique $(-2)$-curve in $X$ corresponding to the $g_2^1$ on $C$; in this case, $\Sigma$ has an ordinary double point as its unique singularity.