Re 2. Sometimes the routers you are chasing are split up. The 'centre' of the unit is where your cavalry attack. If there are no enemy there it can look like they are not attacking. (This is a feature of V1.5..not sure about earlier versions)

I always kill routers in open battle. Gains a little more experience. In a city I might not kill routers and instead end the battle early so I dont have to fight them i the plaza.