In my experience buying and selling homes, condos and land once you are CTC that's basically it - you are good to go. Lenders understand that closing is not a specific date, and I'll let the realtor's on here correct me but I believe the CTC is good for something like 30 days. DO NOT change you credit report (opening cards, buying furniture, etc) before you actually have the keys in hand.
As far as the first mortgage payment, it's "skip a month" after the current month. At closing you'll prepay the balance of the month including prepaying taxes and insurance (escrow) , which is why it's best to close during the last day or two of a month (ther daily amount you pay is much less), then its skip the next month (no payment). So if you close on the 10th of August, you'll "prepay" 21 days then skip September and your payment will be due on October - most lenders will work out a payment date, but mine have always been the 1st of the month - keep in mind that most (in my case all) mortgages have a 15 day grace period meaning you can pay say on the 4th or 10th and not be considered late (daily interest of the principle applies, but your "P&I" payment won't adjust - the payoff will slightly).
Your agent should be able to answer any questions you have - the lender too, but generally they are pretty hands off after the application process.
