I really just love games
Yes, you can always move a block, regardless of whether you move your climber or not.
Remember, moving a block is optional. Moving your climber is also optional.
Yes, you may move your climber up, down, and horizontally.
See the rulebook page 5, under section "Turn Actions."
In Rodney Smith's video he clearly stated you cannot move down. Also the copy of the rules in the new Capstone version do not mention anywhere that you can move down. Did you read this in the rules?
Otherwise all your answers are on point.