When crafting an original recipe, how do brewers determine the length of time at each stage of fermentation? Obviously, several factors weigh in, including the exact composition of the wort, the amount of yeast added, the temperature, etc., but do you use the gravity alone to determine whether it's done?
Furthermore, what are the signs that a beer is done with a secondary fermentation, and that it is ready to be bottled or kegged?
When brewers are experimenting with new recipes, they generally have an expectation of time to complete primary fermentation based on experience with similar recipes. As you hinted, and as pkeading mentioned, gravity readings should be taken to confirm proper attenuation. After three consecutive days with no change in gravity (and assuming the gravity is close to the target FG), the beer can be racked.
Yeast metabolism is a function of temperature, so fermentations in the upper range of a yeast's allowable range will be shorter, as will fermentations with a lower starting gravities.
Generally yeast pitch rate isn't a factor, as most brewers won't deliberately underpitch or overpitch.
Secondary fermentation time is more of an art and less of a science. If during secondary, other flavors are added, such as dry hopping or oak aging, then the additive tends to dictate time in this stage. Generally, 2-3 weeks is typical for homebrewed ales, and lagers are often conditioned longer. Strong beers benefit more from increased time in secondary as well. The best way to tell if a beer is ready for bottling or kegging is to taste a sample.
The best way to tell when fermentation is complete is to take gravity readings. You should be able to calculate what the gravity should be when fermentation is complete, based on the original gravity and the attenuation factor of your yeast.
Then, just take gravity readings every so often (maybe every day, or every other day), and watch how the readings change. If fermentation seems stuck, and the gravity readings are not dropping, even though you aren't close to your target, you may want to consider re-pitching, since that could indicate that the yeast is bad. I have also noticed a stuck fermentation becomes un-stuck after I rack to a secondary vessel; I think it is just restarted by shaking things up, and getting the yeast that had settled out back into the mixture.
I've only brewed Ales, so that's what I can speak to, but I always ferment 10-14 days. The fermentation is usually complete within 3-4, that is to say, the gravity has dropped the expected amount*. However, I've found that after that initial 3-4 days the yeast continues to do a little work. As a result I simply leave it alone for 2 weeks.
Occasionally I will rack to a secondary vessel for extended aging or to add more fermentables, like fruit, but that is always recipe specific. I don't know of a sure rule to figure out the secondary time; it's dependent on the reason for secondary.
*Expected amount is a function of the original gravity and the yeast attenuation, or percentage of sugar the particular yeast strain is supposed to consume.
To hit your second question, secondary fermentation is a subjective stage of the beer. Once it is done in the primary, you have beer. The time in secondary is to get the beer to taste how you want. So, sample the beer, bottle/keg it when you like it.
