💡 Detailed Explanation
The correct choice is 'あげ' (from 'あげる' - to give / perform a favor for others). The grammar pattern '[Verb Te-form] + あげる' indicates that the speaker performs an action as a favor for someone else (in this case, '妹' - younger sister). Sentence translation: 'I bought and gave a beautiful hat to my younger sister for her birthday.' 'もらい' (to receive a favor) and 'くれ' (someone does a favor for me) do not fit the direct giving flow to '妹'. 'やり' is for animals or plants.