Each meter of LED strip draws about 1 Amp - so a one-meter strip will last about 4 hours on 8 AA batteries (3 hours for rechargeables). A two meter strip will be half that.
Yes! We suggest using NiMH rechargeables! They'll do a great job at driving the LED strip.
The easiest way to make it dimmer is to go with a 6xAA battery holder - the lower voltage will still light up the strip but not as much as 8 AA's
Use 6 to 8 C or D cells for extra power. A 12V lantern battery will also do a good job, or a 12V lead acid battery.