What do the icons under the monsters mean in the challenge play?

While I don't have a full list for reference, I just discovered that a "long click" on an opponent will reveal a bunch of details, including a list of their current buffs/debuffs and how many turns they will remain in effect.

The icons next to heroes in the puzzle challenges indicate the buffs and debuffs that they start with or gain during the event. You own heroes will often give buffs to themselves or each other and cast debuffs on the enemy. The enemy will do the same.

You will see icons for Attack Up, Attack Down, Defense Up, Defense Down, as well as others. I don't have a full list unfortunately.