Addendum abilities of instant spells apply while the spell is resolving, not immediately after casting it. If the spell is countered, you don't get the addendum bonus.

If an effect copies a spell with an addendum ability while it's on the stack, the copy wasn't cast at all, so you won't get the addendum bonus.

If the target creature is an illegal target by the time Arrester's Admonition tries to resolve, the spell doesn't resolve. You don't draw a card if you cast it during your main phase.