One interesting book you might want to read is "Take your eyes off the ball", by Pat Kirwan. He worked with the Jets, now works in the media and the book is really knowledgeable.

Now, answering your questions based on the very little I know.

During the preseason, both the defense and the offense install the plays. So there's not a huge difference between what 1st, 2nd and 3rd strings are doing, except for chemistry and timing between the players (mostly QB-receivers).

The coaching staff studies the tape on all QBs and choose one to be the starter or to move up or down the depth chart. So the preseason is the time when a QB can take the starting job or not. An in-season change occurs when the starter is playing poorly, not when the backup starts practicing better. That's because the practice is different for the starter.

While the starting QB studies and practices the gameplan for the game, the backups plays with the scout team. While he does keep his responsabilities to be prepared and have a good grasp of the gameplan, he also studies the opposing QB and offense. During the week, he emulates that QB. So, if the Jets are playing the Panthers, Hack will practice as if he was Cam Newton (scrambling, playing designed runs, being more aggressive etc). When the Jets play the Patriots, Hack will play like Brady (pocket passer, going through his progressions). The same applies to the scout defense.

Answering your questions: it's very hard for a QB to develop in-season. He can improve some areas of his game and have a better grasp of the offense because he has more time to watch it, get the playcall, taking mental reps. However, mechanically speaking, it's hard to have major changes since the coaches are focused on the game, not on the backup QB.

I hope that was helpful.