Learn Swedish: Jag hade tid att rasta hunden åt dig. - I had time to walk the dog for you.

The preposition åt is used when you do a chore or task on someone else's behalf, taking it off their hands. Here, you are doing the physical work of walking the dog so the other person doesn't have to. You would use till if you were physically giving something to someone, and för if you were doing something for their emotional benefit or sake.

Could I use the verb promenera instead of rasta to mean 'walk'?

No, promenera means that you yourself are taking a walk. To walk a dog, you must either use rasta (which specifically means to let an animal out to exercise and relieve itself) or the phrasal verb gå ut med (to go out with).

Why does the sentence use the definite form hunden (the dog) rather than saying 'your dog' (din hund)?

Just like in English, Swedish often uses the definite form when the owner is obvious from the context. Since you are doing this chore åt dig (for you), it is naturally understood that it is their dog. Saying din hund wouldn't be wrong, but hunden sounds more natural in this everyday context.

Is the word att necessary before rasta?