League of Legends – What Triggers the Eye of the Herald Opening?

Bonus health means health gained from items / runes / masteries.

At level 18, Shen naturally has 1958 health*, so Ki strike bonus damage is equal to (Total Health - 1958)*.08.

*Shen suffers from a reduction in health due to the Inverse Ninja Law when Akali, Kennen, or both are on his team. This will reduce the base health of all involved by 1 or 2, giving Shen a range of base health from 1956-1958.

I see no reason why they would do damage, and my experience playing against her is that they stun but don't damage when that ability is used. The ability itself is the damager and they're just a way to manipulate the balls themselves to some end. All of her other abilities have some damage attached to the balls if they are doing damage themselves in the description.

(Active): Conjures a Dark Sphere at a target location, dealing magic damage. The sphere lasts for up to 6 seconds and can be manipulated by Syndra's other abilities. Transcendent Bonus: Deals 15% bonus damage against Champions. Cooldown: 4 seconds

Damages at the summon location

(Active): Throws the grasped Dark Sphere or enemy at a target area. Enemies struck take magic damage and are slowed for 1.5 second.

Explicitly states how the ball damages

(Active): Draws upon Syndra's full cataclysmic power, harnessing all Dark Spheres to deal magic damage to her target per sphere. Unleashed Power will use the three spheres that orbit Syndra, ensuring a minimum damage. All Dark Spheres remain on the ground for 6 seconds after Unleashed Power is used.

Explicit as well with how the balls act.

So it can be assumed that they're being explicit with the Scatter the Weak, if the balls did damage, they'd state it did some damage.