You need to look at the GPS datasheet and verify which edge you should be looking at. Some GPS modules put out a relatively narrow pulse so if you looked at the wrong edge, you could be as much as 999+mS in error. The minimum error you would face by looking at the wrong edge of the GPS PPS signal is 500mS assuming a 50% duty cycle. The time signal provided by a GPS receiver is going to be the most accurate time signal you are likely to ever encounter in every day life. With satellite lock, it's not going to be off by even 1uS, much less a large fraction of a second. It should take less than 1mS to set the clock in the chronodot so you should be that well aligned.
The MSB of the seconds register acts as an oscillator control bit. If it is set, the oscillator is off. The factory default in the chip is for the bit to be set when the device first powers up. If you don't have a backup battery installed, your chip likely powers up with the oscillator off every time.
Don't automatically expect that because the falling edge is important to the clock chip that it is also the important edge coming out of your GPS module's PPS output. That may not be the case.
The DS1307 is sensitive to noise pickup on the crystal if the case is not grounded. It can also pick up environmental noise adding significant error to its timekeeping ability. The added noise generally makes the clock gain time as the internal ripple counter is seeing extra clock ticks. I assume that the chronodot has the same potential problems, but I have never used on. The chronodot is supposed to be very accurate. If you are seeing drift that results in things being visually out of sync after only a few minutes or noticing that it is not keeping time well, then you have a noise pickup problem. For example, picking up 60HZ AC noise will add 2-3 minutes per day.