Pearstone 3.5mm Stereo Male to Female Extension Cable (Black, 10')

Can this transfer phantom power to a microphone without reducing the quality of the sound?

No, phantom power (48V) is typically transferred through an XLR input on a camera, mixer or recorder, often for shotgun style condenser mics, and large and small diaphragm condenser mics. Perhaps you may be thinking of plug-in power, which is a very low voltage (such as 3 to 5V) supplied through the 3.5mm mic input on devices like cameras or recorders for lavalier style condenser mics. The Pearstone cable can transfer plug-in power, but not phantom power.

Is this capable of recording sound from a mic into a sound recorder?

This is a 3.5mm extension cable, if you have a microphone and recorder with a 3.5mm connection you can use this to extend the cable from the mic to the recorder.

Is this cable compatible with Apple earphones?

This will work fine with Apple earbuds and most headphones for audio listening, but will not carry the signal for the control or microphone features.