“Don't try the paranormal until you know what's normal.” - Granny Weatherwax
1. Familiar movement: For a familiar every special terrain is treated as water. This means, that in case lava and pits the familiar can move through, but has to spend double movement points, Besides, can a familiar end its movement in a pit/lava?
Correct. A "normal" familiar treats lava spaces and pit spaces as water. Also it is unaffected by standing on lava (so far, familiar lacks a health value and cannot be defeated).
A different matter for "familiar treated as figures" (they should have come with another term for them). A familiar treated as a figure (so far, only the Necromancer's Reanimate) is affected by terrain as a hero, has a health score, and can be defeated (it is removed from the map).
2. About large monsters and pits: What if a hero jumps down into a pit, and a large monster want to move on top of that pit? It cannot do that, because the space is occupied, or it can stand above the hero?
A hero in the pit is still a figure. He occupies the square he is in (there is no Z-axis in the rules). A hero may be thread on by another figure only when some effect causes the hero figure to be removed from the map and be substituted with a hero token.