10.за

5.63

2. *пробыла у меня _ два часа*. I had her for two hours. There is no need to put anything in.

3. *Мы уже сняли квартиру НА четыре недели*. We rented an apartment for four weeks. For how long? Four weeks.

*чтобы пожить у нас _ месяц*. To stay with us for a month. There is no need for a preposition in this sentence.

4. *но мы с ним разговаривали _ три часа*. The same case as in the previous sentence. They live for a month, they talk for 3 hours.

*мы ЗА три часа обсудили все вопросы*. We discussed everything in three hours. The preposition *in* English makes perfect sense.

5.64

1. *просидела у нас _ целый час*. See past examples. She sat with us for an hour.

2. *где-то гуляла _ полчаса*. The same. ^ She’s been walking around for half an hour somewhere. As you may have noticed, the preposition *for* often appears.

3. *пробыли там _ весь день». The same. ^

6. *Он стал директором всего ЗА полгода*. How long did it take him to become a director? He became director in six months.

8. *Элиза прожила в США _ два месяца*. See the examples above with zero preposition. She lived in the United States *for* two months.

9. *Нам дали текст для перевода НА неделю, и мы его переводили _ всю неделю*. We were given a text to translate for a week, and we translated it for a whole week.

10. *Этот молодой инженер стал директором ЗА три года*. He became director of the plant in three years.

And in general, you did very well! Despite my corrections, the result is very good. I can imagine how difficult this subject is.

Applying to your exercise, the intuitive rule that I can think of is:
1. За — the action takes [time period] to happen. Emphasis is put on the result. Literally “subject uses [time period] to achieve something”.
2. На — the temporary state of the subject is going to last [time period]. Literally, “for [time period, the situation is going to be different from usual”.
3. No preposition is required in constructions like with the meaning “spend [time period] doing something” and similar in sense. Since your exercise book says “dove occorra”, you’ll have some places to leave blank.

Compare на/за (5.64-10):
Ивана в офисе очень ценят — он стал директором всего на полгода (your variant). Translation: Ivan is valued in the office a lot — he has become a director for only half a year. Not really that valued, eh?

Ивана в офисе очень ценят — он стал директором всего за полгода. Translation: Ivan is valued in the office a lot — he has become a director after only half a year.

Compare -/на (5.64-2):
Секретарша ушла из офиса на минутку. Secretary left an office for a minute (and should be back in office after this minite).
Она где-то гуляла полчаса. She spend half an hour walking somewhere.

Compare -/за:
Я планировал приехать домой за тридцать минут, но в итоге простоял в пробке два часа. I planned to arrive home in thirty minutes, but spend two hours in a traffic jam.