Does this boost the damage from Sapling Toss? Does it affect impact damage and explosive ward?

No.

Banner of Command only increases:
* Yorick's Ghouls and Revenant
* Mordekaiser's Clone
* Annie's Tibbers
* Heimerdinger's Turrets
* Elise's Spiderlings
* Malzahar's Voidlings

It may affect Shaco's Hallucinate, but I don't know.

It may also affect Wukong's Decoy and LeBlanc's Mirror Image, but I'm fairly certain Banner of Command only affects auto-attack damage (of which these deal none).

Zyra's Plants, Shaco's Boxes, Maokai's Saplings, etc, are classified as traps/wards. Banner of Command only affects minions.

It does affect LeBlanc's Mirror Image, Wukong's Decoy, and Shaco's Hallucinate.

It's fairly strong on Shaco. Not only does his clone start doing more damage from the aura, he also projects the aura from himself to the minions nearby! The AP also increases the damage enemies take when the clone explodes.

LeBlanc only feels a bonus on her clone if it has items that add damage, e.g. Wit's End, since her clone has innate 0 AD.

Wukong's Decoy benefits from the AP, while Wukong himself appreciates the armor and pushing power.