How does “change direction while moving” upgrade work? How does it apply to “jump”?

Normally movement is linear and you cannot change direction when performing any movement. With the change direction upgrade you could you can change direction with each movement point (following rules for that movement skill). For the jump skill: If you have the change directions upgrade you count spaces as you normally would and land one space beyond. As long as you are counting spaces orthogonally you can essentially jump diagonally when you have the change directions upgrade.

Can i move fewer spaces than my current range allows?

Yes. However you have to move at least one space (in addition to the natural tumble). The only way to NOT move with an activated movement skill is to block the movement against a wall (or board edge). (The Q side is not considered a movement skill.)