I think either 'walk' or 'walking' would both be correct. Like English, the French present tense cover things that are currently happening or happen normally.
The "at night" comes from "la nuit." I think that in French you don't need to say in/at when talking about a day or time of day. Such as "Le matin je bois du cafe" meaning "In the morning I drink coffee" or "Mercredi je vais aller chez mes parents" meaning "On Wednesday, I am going to go to my parents' house"