Newbie need help to make a motor run for a laser scanner and some questions
A servo motor would have a lot more control than a normal motor. You usually command them to go to one of the predefined steps (stops) by using a function call in the servo motor library.
You can run independently of the computer. Many people do this. Certainly you can run off a battery. It doesn't have an ON button, you normally would have this between your battery and the Uno (eg. built into the battery pack). Like this:





A 9 V battery? I wouldn't use one, especially to drive a motor. It won't last long. Get something like 3 x AA batteries in a battery holder.




To show you the idea, I have this servo motor (SG-5010) I got from Adafruit. 

I measured it using about 200 mA of power when operating. Now, I looked up a typical Alkaline 9 V battery, and found that, consuming 200 mA, it lasts for around 400 mAh. Thus, you could say it would be good for two hours at that rate. Plus, the Arduino will take around 50 mA, so maybe less than two hours.


A smallish servo motor won't need a motor shield. The pictures on the linked page show ordinary brushed motors. A servo motor has 3 leads:


Power (eg. 5 V)
Gnd
Data



Make sure you connect the grounds together, but the power could go to a separate power pack. A bit depends on how much power your motor will use. The data goes to the Arduino for the information about which rotation point to step to.

You should be able, in code, to change the rotation position, forwards and backwards.