In the first sentence, it is an example of an anaphora (rhetorical word repetition in the beginning of the phrase).
It is a thing in English as well: "in sickness and in health, in good times and in bad" etc.
It is grammatical. You don't need to use it in this sentence, but you can.
Взрослые вспоминают его с волнением и с нежностью.
Волнение и нежность are parts of the same gamut of emotions and they come close in the sentence, so you can omit the preposition.
The second sentence is a little bit different.
Я люблю ходить в музей и в кино
В музей and в кино both answer the question куда? (not во что?) and they are different places.
Compare the two sentences:
Он заработал состояние на (чём и чём?) шкафчиках и тумбочках // He made a fortune on cabinets and nightstands.
"Cabinets and nightstands" can be thought of as a single object with the preposition на.
— Где ключи от машины? — Я не знаю. Посмотри (где и где?) на шкафчике и на тумбочке. // "Where are the car keys?" "I don't know. Look on the cabinet and on the nightstand".
Each phrase with на answers the question "where" separately. It is stylistically better to add the preposition to every object.