minimum requirements for ATmega chip

Super noob question:

I have a project that takes inputs from a sensor- two analog inputs- and uses the values to give two pulsing outputs to a servo. It also drives five LEDs that turn on when the sensor is giving a reading that is outside a certain boundary. Basically it's like a gimbal- it senses with a 2-axis G sensor and then tells the servos to move.

The issue is the size. The Uno I am using is way too big for my little servo gizmo. Is it possible to just program the ATMega chip, remove it from the Uno, and then put it into a socket on a separate small PCB that has the LEDs on it and the sensor soldered to it? For accurate pulses, do I need some sort of crystal? Oh, I already have a 5V regulated power supply that I assume would power the ATmega chip. I see little capacitors and resistors on the Uno, but they look like they are for the USB translator chip and on board LEDs, but maybe not...

Sorry for the unbelievably noobish question, but I never took an electronics class unfortunately and am trying to teach myself with the help of all you excellent people. Thanks!

Could go with a DIP sized version:

Leave off the parts you're not needing.
Or something a toulch bigger:
http://www.nkcelectronics.com/arduino-runtime-board-rev-b.html
Or a surface mount version that's in between:

I'd predict you would not need a crystal/resonator with your AVR to run the servos stably. Servos don't need a high precision signal.

Your biggest problem will be the servos causing fluctuations on your power line. See De-coupling to get a start on understanding that type of issue. Basically it comes down to adding the right size or combination of capacitors between your 5V and GND lines on your AVR to smooth out the fluctuations. There are no hard rules here; feel free to experiment.

ATMega328 is overkill for this type of problem. I'd look at using an ATTiny85 or 84 for this.

Thanks! I will look at the tiny chips as well.

ATTiny85 only has 6 IO pins.
Stated requirement needed 9.

Need to pick a part with more IO:

Having a serial port makes debugging a lot easier. Don't kill yourself my making it so small (IO limited) that it becomes difficult to debug.