Ok, wow, that's getting pretty complicated. I like that your explanation for the phrases "for help" and "to bake some cookies" is the same (complements) since it looks to me like whatever they do it should be the same in both sentences. From my amateur perspective they seem somewhat adverbial because they kind of answer "why" I asked.
I'm not actually studying linguistics, but just reviewing grammar so I can write better. I live in California and work as an engineer.
If anything, "for help" and "to bake some cookies" answer the question "what" not "why". What did she ask her mother for? answer = "help". And what did she ask her mother to do; answer = "bake some cookies".
A "why" adverbial (or reason adjunct) would be:
I asked my mother for help because I didn't know how to make cookies. I asked my mother to bake some cookies because the larder was empty.
In those examples, the underlined expressions are 'reason' adjuncts (adverbials); they answer the question why I asked my mother ...