When you change the scene, such as entering or exiting a building, the people on the streets will respawn. Fighting the groups will earn you experience, which allows you to unlock combos and upgrades. There are certain side missions that also require you to fight gangs in the streets. Such as protecting the political prisoners, and Riddler's henchmen. You often cannot enter a building unless the surrounding area is free of thugs.
To your specific questions:
- While you are in an area it will remain clear unless you cause the game to reload the area (such as entering and exiting a building).
- You earn experience and uncover Riddler secrets (when apprehending Riddler thugs).
- Since you are flying over rooftops to get to the phones, you can easily circumvent the thugs. Only when you arrive at a payphone will you have to knock out any nearby thugs. Since you don't know which payphone it will be, there is little point in eliminating all the thugs on the streets before pursing one.
Certain buildings once cleared will remain cleared, such as the Church, and the Museum.