A Uno will work fine for what you want. The servos will definitely need an external power supply. Connect the external supply and servo grounds to the Uno ground. A 9V transistor battery will power just the Uno, but will not last very long.
Yes, you can.
You make sure your external powersupply can deliver the voltage (5..7.2?) and enough current for your servos.
Arduino is happy with 9V as input via the barrel power-connector
Ok thank you!
So where do I connect the two wires from the power supply? One to the ground, and the other? Do I still have to plug the 9v battery? From what I understood the answer is yes
all GND common. (servo black line
9V to arduino Vin (or barrel connector)
control lines from digital out to servos (orange line)
7V !! (check datasheet) to servo powerline (red line)
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Usually sevos fancy 5V..7.2V I would not connect 9V to a 7V servo
Find a way to drop down 2V:
prebuilt voltage regulator, transistor+zener, 3 serial diodes, or maybe even a 5-7V power adapter
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It is smart to power the servos from arduinos 5V output. (not enough current)