1. Correct, Hull Down is a state distinct from Standing, so Jumping Movement is not available in that situation.
We probably could benefit from stating that explicitly in TO.
2. A Quad 'Mech can be either Prone or Hull down, but not both.
3. See 2. Further:
"Prone: A ’Mech can move from hull-down to a prone
position without any MP expenditure. To move from a prone
position to a hull-down position requires the expenditure of 1
MP (+1 additional MP for each missing and/or destroyed leg
actuator), but does not require a Piloting Skill Roll"
Finally, you cannot be Hull Down when starting the Jump, nor can you go Hull Down using Jump MP. This latter item should be clarified further in TO.
4. No, Hull Down cannot be achieved or left using Minimum Movement, unless the cost is 1 MP, i.e., when going from Prone to Hull Down.