So you would ask for, i.e. 45 day feasibility period (the time in which your deposit is still refundable) and then if by day 45 you feel confident and can start to move towards a close, then you will ask for a certain amount of time to close, but during that second timeframe, the deposit is now NON refundable.

You will increase the rent and security deposit because of the dog's liability.  

Do you know if your security deposit increase is legal in your market?

If speed is of the essence you could offer 100% security deposit return if they’re out 30 days faster than required.

Someone I know owns section 8 properties, he doesn’t have any issues and the rent gets deposited directly to his account.

You don't actually hold the deposit, from my understanding. 

First month's rent and deposit have been paid. 

That means effective showings, effective applications, checking references, background checks, collecting cash or cashier checks for first & deposit and waiting for them to clear, starting the eviction process immediately if they are a no pay, etc.  

However, you are going to forfeit any Earnest money you have deposited onto the home(more than likely anyways, it depends on the fine specifics of your contract on the offer letter).

If you do add pets to the lease, I HIGHLY recommend a non refundable pet deposit.