1. By the lights of general usage, a modifier cannot be a determiner: they are mutually exclusive categories (the one being, by definition, optional, and the other, typically, obligatory). Articles (a/an/the) are normally classed as determiners.
2. Regarding your analysis of the postmodifer: yes and no. Of course, in her hand is a prepositional phrase, but the way you have underlined things here seems to indicate that you consider it to be a part of the noun phrase, whereas it is clearly functioning as a modifier to the main verb phrase, specifying WHERE she had the thorn.
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