The power supply of 9 volts is adequate. Unless the IC is the LM386N-3 or -4, which are rated at higher power supply and output. Likely, the one on yours is the more common LM386-N1.
The circuit you showed related to the kit has a few components that bleed off some of the source signal (guitar or MP3) to ground. Probably to filter out and "clean up" the signal before the last stage, through the 100uF cap leading to the speaker.
This is a diagram of the circuit I use. There are a number of differences from yours. Note the 220uF electrolytic cap between pin 5 and the speaker, and the 0.1uF disk cap on pin 7. These changes, along with the 10uF cap from pin 1 to 8 (the gain stage), give this amp a well rounded sound. Play your guitar lightly, and it has a nice mid range break up. Play aggressive, and it's crunchy, but not so distorted as to become annoying.
The volume control is more for looks, since a 10k volume pot reacts as either "full volume", or when turned to about 1/2, it is way too quiet. Many builders/players have volume and/or tone pots on the guitar, which allows for better control and adjustment.
As for piezos, I tried them in my earlier builds and they do have merit. But I prefer to wind my own mag pups or use inexpensive ones from recycled guitars.