Actually 너 is even more offensive than 당신 if said to an adult outside of your close circle of friends, or even for a friend if they'are more than a year or two older than you. Saying 너 to an unfamiliar adult is like declaring you're going to treat them like a child. 당신 should be strictly avoided in conversational context. and 너 should be reserved for similar aged friends and children. There are a few ways to say "Is this your dog?". 1. (이거) 네 개야? - to children or close friends who are not older than you. 2. (이거) 자기 개야? - 자기 (your, his, their, etc) are softer than 네 (but not by much). 3. (이거) 기르는/기르시는 개예요? - 기르시는 is like "that you own" or "that you're looking after", so it indirectly means "your" (The subject is not mentioned, but since no one would ask if the dog is their own, it means "your". If you don't include 기르는, the question becomes "Is this a dog?" ). #1 and #2 are for friends or children, and #3 for other adults. There are words such as 선생님, 사모님, 님, etc. that you might use for "you", but none are general enough to be useful. Also the honorifics (like adding 시 to a verb) can help. Honorifics allow you to imply "you" without actually saying it because honorific phrases always refer to the other person. For example, to mean "your company" or "the company you work for", you can say 다니시는 회사. The honorific 시 automatically indicates it is about the listener.

Haha, it is a interesting question. If you "really" want to know who's dog, you can ask "누구 개 에요?". No need to say "You" every time in Korean. For example, if you want to ask someone to eat dinner you can say"Did you eat dinner?" but in Korean say this " 저녁 먹었어요?". is okay. It is very natural. if you say "너는 저녁을 먹었어요?" or "당신은 저녁을 먹었나요?" it is possible but not perfect. I think that is the most different part of Korean and English. Anyway, my answer is no need to say you every time in Korean. I hope it helps you.