For the second example, the expression is “avoir le droit à qqc”. https://www.wordreference.com/fren/avoir
Where does your first example come from ? I don’t see it in the lesson.
I would use ‘choisir .. un cadeau pour qqn’. There are many examples of ‘un cadeau pour qqn’ on French gift vendors’ sites, but I haven’t found any of ‘un cadeau à qqn’, except if required by the verb form eg faire un cadeau à, donner un cadeau à, acheter un cadeau à (in this case - to buy from).
This is not the situation in your example as the verb form is not ‘choisir… à NOUN’, and with a verb, it would be ‘choisir de INFINITIVE’
https://www.lawlessfrench.com/grammar/verbs-with-a/ https://www.lawlessfrench.com/grammar/verbs-with-de/