Let me give an answer to your first question. 

Let $\pi: X \to Y$ be a dominant morphism of smooth projective varieties over $\mathbf C$ with connected fibers; $E$ a prime divisor on $Y$. The **multiplicity** of $\pi$ along $E$ is defined by
$$
m(E) \overset{\text{def}}= \inf\{m_j\}, \quad \pi^\ast(E) = \sum_j m_j D_j, 
$$
and 
$$
\Delta_\pi \overset{\text{def}}= \sum_i \left(1 - \frac1{m(E_i)}\right) E_i
$$
is called the **multiplicity divisor** associated to $\pi$. 

Let $\pi: X \dashrightarrow Y$ be a dominant rational map of smooth projective varieties over $\mathbf C$ with connected fibers. The **Kodaira dimension** of $\pi$, denoted by $\kappa(\pi)$, is defined to be $\inf\{\kappa(Y^\prime, K_{Y^\prime} + \Delta_{\pi^\prime})\}$, where $\pi^\prime: X^\prime \to Y^\prime$ is taken over all dominant morphisms such that there exist birational maps $u: X \dashrightarrow X^\prime$ and $v:Y \dashrightarrow Y^\prime$ satisfying $\pi^\prime \circ u = v \circ \pi$. 

Let $\pi: X \dashrightarrow Y$ be a dominant rational map of smooth projective varieties over $\mathbf C$ with connected fibers. $\pi$ is said to be of **general type** if $\kappa(\pi) = \dim(Y)$. 

Let $X$ be a smooth projective variety over $\mathbf C$. $X$ is said to be **special** if there is no dominant rational map of general type with connected fibers from $X$ to any smooth projective variety $Y$ with $\dim(Y) > 0$.

**Theorem.** If $X$ is special, then the Albanese morphism $\alpha: X \to A$ is dominant with connected fibers and $\Delta_\alpha = 0$. 

Proof. [CAM] Proposition 5.3. 

**Theorem.** If $X$ is rationally connected, then $X$ is special (but of course the Albanese morphism is trivial in this case). 

**Theorem.** If $\kappa(X) = 0$, then $X$ is special. 

**Theorem.** If $-K_X$ is nef, then $X$ is special. 

**Theorem.** For any $n > 0$ and $\kappa \in \{-\infty, 0, \dots, n - 1\}$, there exists a special variety with dimension $n$ and Kodaira dimension $\kappa$. 

[CAM] Frédéric Campana. *Orbifolds, Special Varieties and Classification Theory*.