Tapping the jump button will keep you in the air for longer than just holding it down.
Genji is able to play a variety of roles. He can set up ganks, push droids or focus on supportting with shields and heals.
Genji will stop moving to cast Monarch Blessing. This can be annoying or even dangerous depending on where you do it, due to his poor acceleration. His movement will not be interrupted if Monarch Blessing is cast while he's in the air, however, making it a safer option.
While inside Cocoon, enemies are invulnerable to all sorts of external damage. Tell your allies to save their abilities for when the target comes out of it.
Cocoon will bounce off of walls and ceilings until it hits the ground, a target, or it disappears.
Cocoon is one of the most versatile abilities in the game. It can be used to secure a kill, allowing your teammates time to proper position themselves against the victim, as an escape mechanism or (with the proper upgrades) as a powerful pushing tool. Experiment with different upgrades to find your playstyle.