looks like a very crude and very inaccurate D-to-A converter and a crude two-pole filter feeding the LM386 through a 1k-ohm resistor.

Yes, I'd prefer a separate (9-12 volts) power supply, and the link between the LM386 and the digital circuit + filter should be capacitor coupled to the LM386, rather than to the speaker, though I guess the cap to the speaker is tolerable if it is MUCH larger. The best this will sound is bordering on terrible, though, so don't expect too much. With a little attention to detail, it can be improved.

Yes, 10K/20K values would probably be more accurate, though increasing the 33K rsistances to 34K by adding a 1K in series would fix it, too, and it would have been apparent to you that a 0.1 uF cap on the output of the amp is WAY too small, had you looked at the LM386 datasheet.

http://cache.national.com/ds/LM/LM386.pdf

Most of the examples use 250 uF, though I noticed one oscillator that used only 50 uF. They tend to have 0.1 uF as a bypass to GND on the output.

Since your rather small speaker is likely to have poor low-end response, I'd recommend you consider the example labeled Amplifier with Bass Boost. You might also consider the gain control between pins 1 and 8. Further, the use of a potentiometer, the wiper of which goes to the input of the amplifier is probably advisable as well.

I'd suggest you start by reading the application notes in the datasheet once again.

RE