yes, 我的狗 is correct

Your sample is correct. 地+verb 的+noun

you can simply see "我的" as my "他的" as "His" It is that simple.

的can be used in many many situations one is between the subject and object, (if the object belongs to the subject) examples：我的狗，我的花 but “的”can also used in this situation：an adjective+的+subject to describe sth 开心的笑容 美丽的花 春天的花