1. Перелейте молоко из бутылки в кастрюлю.

It is the only option, since it implies "pour the milk from the bottle to the pan". The verb "перелить" assumes the whole amount of the liquid should be poured from one vessel to another. The same would be for any other liquid:

Перелейте воду из А в Б. Перелейте вино из А в Б. etc.

If you meant one should pour a part of the liquid (not the whole amount), a different verb would be used, "отлить":

Отлейте молока (воды, вина etc.) из бутылки в кастрюлю. - Pour some milk (water, wine...) from the bottle to the pan.

Then again, I would say "передайте воды" because some kind of amount (a bottle, a glass etc) of water is implied. Is that the case here as well? What's the rule for this?

If there is only one bottle of water on the table, I would ask: "передайте воду" (pass me the water) if the whole amount is assumed.

If there is an unlimited source of water, and I need to have some, I would prefer: "передайте воды" (pass me some water). But "передайте воду" is still possible. I would say, the accusative case is more like universal in this kind of occasions. The partitive (a subtype of genitive) is more limited. So, technically, the accusative is possible everywhere, but there are some cases where the partitive is preferred.

To summarize it,

Rule I. If the whole amount of a substance is assumed, use Accusative (e.g. "pour the water").
Rule II. If a limited amount of a substance is assumed, use Genitive or Accusative (e.g. "pour some water"). In many cases, the genitive would be preferred, but sometimes there can be some restrictions when the genitive is not possible. If in doubt, use the accusative.

2. Подлейте мне, пожалуйста, ещё молоко/молока в чашку.
- I would go with genitive because of the word ещё.

I agree. But I would add some more thought:

In this sentence, "ещё" can have two different meanings. They are quite similar, but there are some subtle nuances.

a) "ещё" can be related to "молоко" (meaning "more milk"). In this case, you are right: only "ещё молока" is possible ("ещё" specifies an amount, and it can be replaced with "больше" (more), "побольше" (a bit more)):

Подлейте мне, пожалуйста, ещё молока в чашку. Подлейте мне, пожалуйста, больше молока в чашку. Подлейте мне, пожалуйста, побольше молока в чашку. = Please pour some more milk to my cup.

However, please note: the usage of "ещё" and "больше/побольше" is not interchangeable. "Ещё" means you just want to have some more milk in addition to what you already have. "Больше/побольше" is literally "more", it means you are definitely not satisfied with the amount you have, it is not enough, and you say you need a larger amount.

б) "ещё" can be related to the verb "подлейте" (meaning "pour me some milk once more"). It implies the person who you ask has already poured you some milk, and you request him/her to repeat the action. In this case, "ещё" can be replaced with "ещё раз" (once again) or "снова" (again).

In case of б), rule II above is applied: genitive is preferred, but accusative is still possible.

If someone says "Подлейте мне, пожалуйста, ещё молоко в чашку", I do not think it would be considered an error. But maybe it sounds a bit clumsy. And it can only mean "Please pour some milk to my cup once again".

3. Положите себе в чай варенье/варенья.
- I'm told варенье, but I'm not sure why.

See rule II.

4. Продавец отвесил девушке муку/муки.
- Муки, because a certain amount of flour, say a kilogram, is implicitly understood here, right?

See rule II.

5. В борщ добавили соль/соли, и он стал вкуснее.
- Not sure

See rule II.

6. Стало легче дышать, будто кислород/кислорода вдохнули.
- No idea.

Here, I would use "кислорода" to emphasize it is in addition to what we breathed in before. Because actually it assumes "ещё кислорода" (some more oxygen).

The option with accusative (стало легче дышать, будто кислород вдохнули) sounds a bit strange, as if you did not breathe any oxygen before, which is impossible for a human being.

7. Мы принесли из магазина хлеб/хлеба.
- Хлеб?

See rule II.

Rule I would be only applied if you bring the whole amount of bread which was in the shop.

8. Я не понял вопрос/вопроса.
- Genitive because of the negation?

Yes. But it is optional as well. The genitive of negation is similar to the partitive genitive - they both have some restrictions, and both can be replaced with accusative in most of cases.

Warning: not to be confused with the genitive of absence! У меня нет вопросов. - The genitive is the only option here.