It's rather simple.
You can see the players ratings from previous seasons by looking at the "Spring Training" lines of his profile. Look from one season to the next, and the next, and the next, etc. Then you also have the ratings from the current season, starting with "Opeing Day" and all the "Regular Season" updates. You'll notice a pattern. After a while they generally slow down and eventually come to a halt (or near halt).
Looking at all these in total, it's fairly easy to see when a player has stopped progressing, or that he's just about there.
As to your other questions: ratings can be grouped into two categories: physical ratings, and skill ratings. Physical ratings are a factor of age, training budget, and player makeup. Skill ratings are a factor of coaching and playing time.
In general: physical ratings will tend to peak and then start to decline at a different time than the skill ratings. So you potentially see some players who start to decline in some areas (power, speed, range, etc), while staying steady or even increasing in other ratings (contact, splits, etc.).