Like Karlshammar says, you don't have to provide proof unless they ask. You have to list all periods you lived in Canada and all periods you didn't. If they verify this information, for example by getting data from the US border and see that you have lied or they are just suspicious of your information, they may ask you to prove that you are telling the truth. Proof of having lived in Canada, I would think the best one would be bank records. If you can go to your bank and get back statements (unless you still have your statements), you can show that you were taking out cash in Canadian ATM's and shopping in Canada so obviously at that time, you must have been in Canada.
You can leave Canada after you have applied for citizenship but you need to keep them updated on your address or keep an address in Canada because when they send you the letter to take the test, you will not have very much notice, maybe 2 weeks and you have to be ready to drop everything and go. Yes, you can postpone the test but that means they will schedule a new one, who knows when, could be a year later and it might not be a convenient time for you either. Same with the ceremony 2-4 months after the test, they will send you a letter, you will know about 2 weeks in advance when the ceremony is. Be there. They will not let you schedule these dates when it is convenient for you.