Let's 101 for a minute! I'm glad you asked this, hopefully we can help you and anyone else reading this!
-Never, ever, ever run a balance on a credit card. Not for points, not for promos, not for anything. Running a balance past the due date will become a part of your credit report.
-Read your fine print. There will be fine print describing exactly how the rewards program works. As for general info what this rewards fine print should include is when and how you get rewards. If they say end of the billing cycle you should see your rewards within a couple days of the billing cycle ending.
-Some people pay early. This is usually for peace of mind, you can pay even before the statement is issued. Heck, I was so scared of messing things up with credit cards in college that I paid every charge as soon as it posted! (Obviously, I was scared, it was for peace of mind and I still got all my rewards points). I don't do that anymore but still early payment does not affect your ability to qualify under any credit card rewards program I've ever seen.
As for this specific credit card if the info is not on your online account with Chase, call the number on the back of your card if you're wondering when your statement closes (if you want to know your due date they should be able to tell you that as well).
Disney Rewards are, like many bonus programs, their own annoying pain in the rear program to get your points out of. Points post when your billing cycle closes and then you can* transfer them to your Rewards Redemption card.
*You have to have $20 worth for your first redemption, before you can even order a redemption card actually
*After the initial redemption, you can load more $$ onto the card as soon as you have built up at least $10 worth of Rewards
Do note, if you are close to traveling to Disney and your redemption card doesn't come in time, or if you forget your rewards redemption card at home, you can pick one up at various locations in DLR and WDW (think like city halls) and then call Chase and they will transfer your rewards balance to that card for you to use on your trip.