The inclusion of an open affine, as well as the map $\mathbb{A}^1\sqcup \mathbb{A}^1 \to \mathbb{P}^1$ are flat. The thing is that the theorem of Ehresmann applies only when the map is proper.

There are different ways to make intermediate Jacobians. But as far as I know, either they produce abelian varieties, that is they come with a polarization (the Weil ones) or deform holomorphically in families (the Griffiths ones), but not both at the same time.