If the creature you control is an illegal target as Signature Slam tries to resolve but the creature you don't control is still a legal target, each modified creature you control (which may include the target creature you control if it's still on the battlefield) will still deal damage equal to its power to the creature you don't control.

If the creature you don't control is an illegal target as Signature Slam tries to resolve but the creature you control is a legal target, you'll still put a +1/+1 counter on the target creature you control.

An Aura controlled by another player does not cause a creature you control to be modified.

A creature with a counter on it is considered modified no matter what kind of counter it is or which player put it on that creature.

A creature that is equipped is considered modified no matter who controls the Equipment that's attached to it.