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.

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

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.  

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

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.