In my mind, it depends on whether the plural of "toys" means "many toys of one type/category," in which case it would be the first, or, "many toys of multiple types/categories," in which case it would be the second.
If there were multiple addictions to different, non-interchangeable types of toys, the plural "addictions" would be proper, but in any case, I think the singular "addiction" reads better and doesn't confuse the sense of the sentence.
Yes, the first sentence is not correct in any situation.
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