I am new to Airbnb and didn't want to do the auto book because on homeaway, many times guests will see that we don't allow pets and then when we speak to them, they ask if they can take a small dog, etc. or we put max 8 guests and then they will tell us that they have 12 in the party. Will that really move our listing down so that we won't get bookings?
Yes, if you have instant booking (IB) off, it will push your listing down. However, many hosts that don't use IB and still manage to get bookings.
If you decide to turn it on and someone books but they will break the house rules (more guests than allowed, bring pets, etc), you should be able to call AB and cancel penalty free. Some host were also able to cancel penalty free if they said they were uncomfortable with the booking.