This brings us to topic of Affinage. How does one tell how long is right?
By taste using a trier, or by projection of the batch based on past knowledge of aging times for the cheese style. Alternatively, you can be more exact about it and measure the degree of protein degradation in the final cheese and pull to sell when a certain percentage has been broken down by cell proteases.
How to know if a cheese can continue to improve with more age?
This is a somewhat relative measure because improvement may result in more intense flavors, but if consumers are not used to it, sales will be tough.
The first indicator is calcium, though, which is determined by whey drain pH. A high calcium cheese has the theoretical structure necessary to age. Calcium is like the framing in a house.
Second indicator is moisture in non-fat. Moisture has a lot to do with rate of maturation. Low moisture cheeses take longer and mature more slowly. A high moisture (above ~38%) usually will not last more than a year unless you store at cold temps.
Beyond that, you taste it. With experience, you can tell when a cheese will age well. It's like with winemaking. I know guys who will go out to the vineyard and taste the grape and tell to the half day when acidity and Brix are perfectly balanced for the varietal. Then they can taste green wine and know how to blend it to achieve a 3-year, 5-year and N year drinkeability.
When you taste cheese for long term aging, what you're looking for underlying body and rate of flavor development. After 60 days, a cheese with extreme or moderate flavor usually does not age well. A cheese with some flavor and inklings of aroma ages slower and will be better over time.
Finally, there are the theoretical expectations for how long aging takes based on the culture selection.
I realize that there are years of experience to get this part really right, but where do we start looking?
Hope that helps. Start by keeping a notebook and tasting cheeses at various points in their life, and writing down what you smell, taste, feel, and sense. Over time, you'll know how to spot good cheeses with aging potential based on how they are young. Have to train the palate.