Deeprun Rats are one of the aggressive critters that will fight back if they survive an attack. They do minimal damage. With only one hit point, you have to attack them with a non-damaging attack to see this behavior. Priest's Shadow Word: Pain has a delayed damage, and if cast on a close Deeprun Rat, the rat will have time to attack once or twice before it dies. Mages can use polymorph.
Enthralled rats turn Alliance-friendly (maybe not coded that way - they turn friendly to the flute player's faction, and only Alliance characters can get the quest) and follow the flute user around like small pet companions. This is one of several ways to get multiple companion critters, and can last indefinitely (until you log off, time out, or the server shuts down for maintenance), but you must remain within the Deeprun Tram system. When you exit the Deeprun Tram instance, the Deeprun Rats disappear.
The Deeprun Rat Roundup only requires you to have five Deeprun Rats enthralled when you turn the quest in, but you can use the flute to enthrall far more; they will all follow you around. They all disappear when you turn the quest in, and you get nothing extra.
The Rat Catcher's Flute disappears when you turn the quest in, so the only time you can tame the Deeprun Rats is during the quest.