Was the Earth-Sun distance ever calculated based on the time difference between night and day? If we define sunrise and sunset as the time when the center of the Sun goes over the horizon then day time is a little bit shorter than night time because from a single point one cannot see a complete hemisphere. Computing backwards from a known Earth radius of around 6700km and a known Earth-Sun distance of around 150 million kilometers one can compute that the angle of the Sun shining on the Earth is 13,4/150.000 which comes to the day being roughly 5 seconds shorter than the night. 

The radius of the Earth was already known to the ancient Greeks. You also need a clock that is accurate to at less than a second per day and a spot to observe the Sun rise and and Sun set at exact times. If you are not at the equator there will be differences in day time due to the tilted Earth axis but one could either measure directly at the equator or measure over a whole year and average it out.

Question: was this method ever used to estimate the distance between the Earth and the Sun?
You have a good idea. You mentioned 4 effects already (clock accuracy, latitude, time of year, and clear horizon), but there is another effect that is larger than those: atmospheric refraction. Refraction causes a rising object to appear to be half a degree higher than reality. Refraction depends on atmospheric pressure and temperature, so it may be harder to compensate for it.
This article on Sky &amp; Telescope indicates that the amount of refraction is not well understood: We Don’t Really Know When the Sun Rises

But Michigan Tech dissertation work by Teresa Wilson (now at the U.S. Naval Observatory) suggests that our estimates are often off by 1 to 5 minutes.
She found that, overall, predicted times varied in accuracy by location and season, with sunrise times over land generally being early in the summer months and late in the winter. Summer showed the largest discrepancies, probably due to the pronounced refractive effect that the large temperature difference in the atmosphere has during those months. Mirage effects due to cold air topped by warm over water horizons also exacerbated lags in sunset times throughout the year, sometimes by up to 5 minutes.