You don't have to choose a target for Absolving Lammasu's last ability. However, if you do, and that permanent is an illegal target at the time the ability tries to resolve, it won't resolve and none of its effects will happen. You won't gain life.

When an effect suspects a creature, it becomes suspected. It gains menace and "This creature can't block" for as long as it's suspected. It stays suspected until it leaves the battlefield or another effect causes it to no longer be suspected.

If a suspected creature loses all abilities, it will lose menace and "This creature can't block", but it won't stop being suspected.

Being suspected isn't a copiable value. If a permanent becomes a copy of a suspected creature, it won't be suspected.

If a creature is already suspected, suspecting it again won't have any effect.

There's no limit to the number of creatures that can be suspected simultaneously. Suspecting a new creature doesn't cause other creatures to stop being suspected.