Though, there is not much information about your servos let's assume that they consume at least 1A each. This value will probably be higher given that you want to move doors and that requires torque.
Now let's see the RPi's capabilities. An answer from Stack Exchange Raspberry Pi, tells you the necessary information and sources about the RPi's power rails and GPIOs.
The 3.3 V supply can source a maximum of 50 mA, as the GPIOs are driven from this rail too. A GPIO can source max 16 mA, the overall current sourced by GPIOs cannot exceed the 3.3 V rail's max 50 mA.
This is far not enough but also 3.3 V would be too low for supplying a servo, as servos usually need 4-6 V.
The 5 V supply rail is limited by the power supply that powers the RPi, minus the RPi's current consumption. You did not provide what model do you use, so here is a table with each (except RPi3) models power requirements.
Now, if you want to power your servos from the RPi's 5 V rail, your maximum current is at least 5A + RPi's consumption. If you check, most USB micro type B connectors are rated on 1.8 A, but even if the connector can handle 5-6 A, driving this much current through the board just to supply your servos would be risky.
The best solution would be an external supply which provides 6 V as your servos will have higher torque at 6 V. Also it must be able to supply six times a servo's current.
A possible block diagram:
The driving circuit can be a simple current limiting resistor an each pin of the RPi or you can use an appropriate small-signal FET as an interface.
As for connecting the LEDs, you can use a simple current limiting resistor and GPIOs, but be careful to not exceed the 50 mA of the 3.3 V rail.