Customizing Weapons for heros?

Does customizing a weapon work on a hero?
Does the Hero get the 20% applied damage, or does that just apply to the character your using it on?

Yes, the hero still gets the +20% damage.

Just to clarify - you don't customize a weapon for a hero, your heroes can use weapons customized for you and they do get the +20%.

if you give a weapon/armor to a hero and then logout. Do the weapons/armor vanish or do they stay with them?

The weapons stay with them. You also can use unused heroes as a storage location for weapons.

The only way the weapons on those Heroes 'disappear' is if you delete your character and forget to remove their weapon sets and place them in your xunlai.