Some more details about -지(요).
The -지 ending form has these major usages.
1. RELATE something factual in a soft register, as if confirming it to oneself.
2. ASK a question with a relaxed tone, either about something one doesn't know or to confirm what one knows.
3. SUGGEST an action in a semi-question form, like "shall we ...?", or "maybe you could ...", for the listener or for both.
4. express a REGRET that the listener has failed to do the desired thing, by mentioning what he should have done.
5. express a SARCASM, like "I don't think much of it but go ahead and show me if you want" (rarer than the others)
#3 usage is similar to -는게 어때요? or -ㅂ시다 as you mentioned.
추운데 문을 닫지요? is a little vague - it can be a request or asking permission so the speaker can do it himself.
It is much softer and polite than 닫아주세요 which is a very direct request, almost sounding like an order.
(닫아 주시겠어요?, 닫아 주실 수 있으세요?, and 닫았으면 좋겠는데 괜찮을까요? are other polite ways to say it)
Here's your examples marked with which of the four meanings above applies.
- 저녁을 안 먹었으면 우리와 같이 /오래 걸었으니 우리 잠시 쉬어가지요(?) (suggest)
- 추운데 문을 닫지요(?) (suggest)
- 내일 오시지요(?) (ask, "suggest" is also possible - context dependent)
- 피터가 몇 시에 돌아오지요? (ask)
- 네, 내일 떠나지요. (relate, "suggest" is also possible - context dependent)
- 저 사람이 누구지요? (ask)
- 이거 얼마지요? (ask)
- 가: 남편과는 어떻게 만나셨어요? (NA)
- 나: 저는 남편을 교회에서 처음 만났지요. (relate)
(I think -지요 is less often used to answer a specific question like this)
- 여기 가까운 화장실이 어디지요? (ask)
* 불꽃놀이는 벌써 끝나는데 조금만 일찍 오시지요. (regret) (끝났는데 seems to make more sense than 끝나는데))
* 아침에 비 온다고 했을 때 미리 우산을 챙기지. (regret)
- 그 사람이 어디서 살고 있죠? (ask)
The phrases expressing "regret" can be considered a short form of a fuller sentence.
- 불꽃놀이는 벌써 끝났는데 (... regret ...), 조금만 더 일찍 오시지 (않고 왜 이렇게 늦게 오셨어요?).
- 아침에 비 온다고 했을 때 미리 우산을 챙기지 (않고 왜 그냥 나갔어?).