I would prefer to say " Я сделала ему подарок" in order to avoid repetition of isonyms.

You can say я подарила ему часы (a particular present)

Yes, it is correct)

Ещё подарок можно вручить. "Я вручила ему подарок". "Вручать" is something between "giving" as action ( "давать") and "giving a gift" ( "дарить"). It usually used in description of awarding process ("вручение олимпийских медалей") but also sounds good in this context.