"First off, with the AC turned off on both thermostats, the furnace fan continues to operate and circulate air. Is this normal to have the furnace unit fan not controlled by the * "
Check the thermostat setting for FAN. Is it set to AUTO or * If it's set to ON, change it to AUTO. If that doesn't help you may have a fan control relay that is defective.
"Second, the furnace is making a high pitch sound throughout the duct work, which to me sounds like the fan might be dying."
It is possible the motor bearings are going bad. Only way to know is check the motor directly, once you stop it.
"Finally, there is a small puddle of water around the unit which may or may not be released to fan sound. Also, the PVC pipe that vents air is full of water. "
The symptoms are those of a clogged evaporator coil drip pan waste line. Find where that PVC pipe exits the house and using a shopvac try and suck any material out that could be preventing the water flowing from the drip pan.
Make and model number of all equipment is always helpful.