yes, turning the feed rate up will increase welding current. The power switches deal with the voltage. Mig is constant voltage as opposed to arc and tig being constant current. Wire size alters the current output too. Arc length will affect current a bit too
Think about it like this... If you increase the wire speed, you are getting on average more metal into the weld pool. That extra metal will carry more current. So yes, increasing the wire speed increases the current. Voltage is set by the switches on the unit and does not vary (much) with wire speed.