If Convenient Target leaves the battlefield, the creature it was enchanting will still be suspected until that creature leaves the battlefield or another effect causes it to no longer be suspected.

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.