Ohh, my...It sounds really weird to say "삼가 고인의 명복을 빕니다." in Korean, when someone lost her dog. I don't think people usually say that way, however heartbroken the dog owner is. I suggest " 심심한 위로의 말씀을 드립니다." or just " 마음이 많이 아프시겠어요.", "너무 상심하지 마세요." Actually, it sounds strange itself to say in a formal way in that case(family dog died)

Yes. it's good expression. or you can say 삼가 고인의 명복을 빕니다. it's means "I carefully pray for the repose of the deceased's soul"

Oh. it was dog. Sorry. 고인 means a person who is died. I think it's not a just normal case, so I think some sentence are better like "I know you lost your family member, How can I say anything about that?", 가족을 잃었는데 내가 뭐라고 할 수 있겠어요? "But I think she or he was happy and still happy because of your family" 하지만 저는 그 아이가 그쪽 가족 덕분에 행복했고 지금도 행복할거라고 생각해요. "she or he is still in your family's heart" cheer up 여전히 가족들 마음속에 있잖아요. 힘내세요. And you can say just 삼가 명복을 빕니다 in this situation no subject(고인) so this expression is used generally. and you can just use you wrote '진심으로 애도의 말씀을 드립니다.'