Word order is flexible for emphasis. Placing köpeği first topicalizes the dog (“as for the dog”), and putting ben right before the verb makes “I” the focus: “It’s I who will walk the dog.” Variants:
Not by itself. Köpeği here is accusative “the dog.” To say “your dog” as the object, you need the possessive + accusative: köpeğini (e.g., Sen dışarıdayken köpeğini ben gezdiririm).
Note: köpeği can also mean “his/her dog” in other contexts, but as a direct object with no possessor shown, it’s understood as “the (specific) dog.”
Gezdiririm is the aorist (general/habitual) and also used for promises/commitments and scheduled/expected actions. It fits the sense “I’ll (be the one to) walk the dog” or “I generally do it when you’re out.”
Gezmek = to stroll/wander (intransitive, you yourself stroll).
Gezdir- is the causative of gezmek: “to make someone/something stroll” → “to take (a dog, a child) for a walk.”
Yürümek = to walk (intransitive). With a dog, you don’t say the dog “walks” you; you “make it walk,” hence gezdir-. Alternatives: dolaştırmak, or the phrase (köpeği) yürüyüşe çıkarmak.
-ken/-yken is tenseless. The time reference comes from context and the main verb. You can say: