You can power your logic and servos from the same battery or different batteries. Just don't power your servos from the same regulator- connect your other electronics to the regulator and connect your servos straight to the battery (if it is 5-7V). If your battery is higher than 5-7V, you need to use separate (high efficiency) regulator to step down the voltage from your batteries for the servos.
Just do not use the same regulator to power your other electronics and servos. Your electronics need a regulator, but the motor inside the servo does not need a regulator and just runs off of whatever voltage you plug into the servo. Most of the power you plug into the servo goes directly to the motor. Some of it is split off inside the servo to a regulator to provide power for the servo electronics so you do not need to worry about that. If you plug the servo into the same regulator as your other electronics, it means you just plugged the noisy motor inside the servo straight into the same power that is powering your electronics.
If this is what you meant to say:
"I can use a seperate supply from a 7805 supply [for the servo]" and assuming the board of education has a regulator built on it, then yes this would work. This is the same as plugging your servo and regulator directly to the battery, and then plugging your electronics to the regulator.
Your second idea is not good. Because, assuming the BOE has a regulator for the electronics, what you just did is you are powering the servo and the electronics off of the same regulator. THe noisy motor inside the servo will likley disrupt everything- and chances are the BOE regulator cannot handle the servo current anyways. PLug the servo straight into the battery, and not the regulator powering the electronics.