If there is no other legal target for the spell, this does not change the target.

You can choose to make a spell on the stack target this spell (if such a target choice would be legal had the spell been cast while this spell was on the stack). The new target for the deflected spell is not chosen until this spell resolves. This spell is still on the stack when new targets are selected for the spell.

You can't make a spell which is on the stack target itself.

Once the spell resolves, the new target is considered to be targeted by the deflected spell. This will trigger any effects which trigger on being targeted.

The target of a spell that targets another spell on the stack can be changed to any other spell on the stack, even if the new target will resolve before the spell does.

This does not check if the current target is legal. It just checks if the spell has a single target.

This only targets the spell being changed, not the original or new target of the spell it is affecting.

You choose the spell to target on announcement, but you pick the new target for that spell on resolution.

If a spell targets multiple things, you can't target it with Imp's Mischief, even if all but one of those targets have become illegal.

You don't choose the new target for the spell until Imp's Mischief resolves. You must change the target if possible. However, you can't change the target to an illegal target. If there are no legal targets to choose from, the target isn't changed. It doesn't matter if the original target has somehow become illegal itself.