Most API's don't do much processing. It's best done in hardware or in a device specific driver anyway. There's too much variety in hardware to create the one true processing method. That being said, at the application level most joysticks will need to be processed the same way just to different degrees.
Most modern joysticks have little to no noise in the analog sticks so filtering is unnecessary. If your stick does have noise, however, lowpass and moving average filtering are a blunt instrument that will make the input feel soggy/laggy. Your best bet is a Kalman filter which adapts the filter strength based on motion and predicted motion. Here is a demo.
Square vs circle depends on the stick. Circle is a safe default but the correct deadzone exactly matches the shape of the slop in the stick.
A different style is independent axes, where you deadzone each axis independently. Use this when you're controlling unrelated things like x-axis turning with y-axis movement.
A consideration, based on your game, is whether or not to normalize the position so that the rim of the deadzone has a magnitude of 0. If you don't then the player won't be able to make a movement slower than the size of the deadzone.
Don't bias based on resting position. This is what the deadzone is for. The stick will rest in a different position every time the player lets go.
Something you didn't mention is shaping which adjusts the sensitivity. You'll want to do this after deadzoning (if at all). A simple method is to use the pow function like so:
x = pow(x, 1.5) To make the joystick less sensitive, for sniping for example, use a power > 1. To make them more sensitive use a power <1 but >0.