What is making this LM386 circuit so distorted?
Input signal voltage can be too high. Max input voltage for the chip is +/- 0.4V before it is damaged and phone output could easily exceed that, so you should use volume control pot at input. Power supply can be too weak, especially if it is a 9V battery, so it does not provide enough current.
No, it does not make any difference
The values look fine. Datasheets usually tells example circuits and why each part is necessary and what it does and effects of changing values are sometimes documented. This is an audio amplifier, changing values by a reasonable amount will have little performance effect, and frankly the part is so old that it specifies capacitor values that are hard to find these days.
C3 at the bypass pin is a bypass capacitor to filter noise and ripple from the internal bias node it connects to. Which also reads in the datasheet.