Vault Kick can be used to jump over walls, or put distance between you and your pursuers. If the enemy champion is in melee range, use Vault Kick to knock them back. If you are already a moderate distance away, use Vault Kick to dash forward even further.
If Robin's basic attack deals the killing blow, This Guy! will not go on cooldown.
Vault Kick will always apply This Guy! whether his passive is off cooldown or not. It could be applied to a different target.
Vault Kick will trigger This Guy! that was applied on a target with a basic attack, and vice versa, but it will not be reapplied. However, This Guy! will still go on cooldown when a basic attack is used to trigger the mark, so avoid doing that.
Dynamic Duo can be used a friendly drone, making it move faster and heal a small amount with every hit it does, although they still can't trigger This Guy! and it is much better to use it on a champion or on himself for the increased lifesteal (Robin does get the other benefits if he targets only himself).