The copies String of Disappearances creates are created on the stack, so they’re not cast. Abilities that trigger when a player casts a spell won’t trigger.

While resolving String of Disappearances, you can’t pay multiple times to copy it multiple times. It’s a string, not a web.

The copy of String of Disappearances has the ability to copy itself. The string can go on or not, as long as players are able and willing to keep paying .

If the target creature is an illegal target by the time String of Disappearances tries to resolve, the spell doesn’t resolve. That creature’s controller can’t pay to copy String of Disappearances.