Say you put Damage Shield on a fig that has Mystics TA (eg. Animal Master) and a character with Toughness makes a close combat attack against them. The DS damage can be reduced, the Mystics damage cannot. In what order does the damage get dealt?

They're almost simultaneous as can be. Bearing that, the Damage Shield triggers when the attack is successful (doing some damage, but let's ignore that for a second) and Mystics triggers when it does damage. So, assumption that the attack's success triggers the Damage Shield. In the case below that is reduced, but then the damage triggers the Mystics retaliation which deals 1 damage even through the damage reducer.

Thanks. I was hoping that the Mystics damage would trigger first and potentially knock the attacker off damage reducers for the DS damage to take effect. Alas, that is not to be.

Actually, check the order of which thing occur stickied above to see why this is so tricky to figure out. Which may reveal that I'm wrong, but it was questioned in that very thread too.

Whenever considering on using the Damage Shield feat, I always try to see if I have room to add on Armor Piercing as well.

But the timing is Damage Shield, which triggers whenever an attack succeeds (damage dealt isn't necessary) which comes before Mystics which triggers when character takes damage from the attack.

QUESTION # 2

What if the Flash - (KC) hits a figure who has a Mystics (TA) and the card "Damage Shield" placed on him .
Is the Flash required to take 2 clicks of damage ?

Yes. The two sources of damage have nothing to do with each other.