When i lose a province, may i receive it back in a future game during this campaign?
I think yes, but read this discussion:
When i lose an achievement, may i buy it back in a future game during this campaign?
When events ask me to pay resources, must i pay all at once or can i pay a bit every round of the game?
I think it's up to you. There are no clear instructions on this. The rules only state that when you pay your Empire, then put the resources on the Empire sheet. My understanding is that you can put resources there during the entire game and at the end, there should be a certain amount of them.
Some events make you lose Production permanentely, if i have 0 production of that type on the empire sheet, how should i interact with this penalty?
Then you lose nothing