Disclaimer. I don't know much about the things I'm asking. This is why my other question pencils on varieties of general type was a bit unclear. I believe the following question makes up for this.

Let $X$ be a smooth projective complex algebraic variety with $K_X$ ample. When does there exist a flat projective (non-isotrivial) morphism $X\to \mathbf{P}^1$ with geometrically connected fibres which are not of general type?

The answer is never when $\dim X \leq 2$. Is the answer also never when $\dim X =3$?

If you assume $\Omega^1_X$ to be ample, then the answer is also never. In fact, in this case, every subvariety of $X$ is of general type.

What if we also put a restriction on the Kodaira dimension of the fibration $X\to \mathbf P^1$, say, the Kodaira dimension is not positive.

What if we replace $\mathbf P^1$ by a smooth projective curve $C$ of positive genus?(This is not the same question, because I want the fibres of my fibration to be connected.) Of course, this is asking for much more. For instance, the Albanese of such an $X$ is going to be non-zero.

Here's a slightly different question: are there $X$ as above with infinitely many distinct abelian varieties $A_1,A_2,\ldots$ mapping non-constantly to $X$? (This is related to "hyperbolicity" properties of $X$.)