Unstable voltages in boost circuit
The voltage drop is because the power source is insufficient to operate everything at the same time.

Amplifier. no, it's not the normal method, but for low power systems it works well enough.  you only get one quarter of the output audio power from the speaker that could have got with a half-bridge drive, which may mean that you are using a more expensive speaker than you would otherwise need. and also you are using twice as much power as would be needed with a push-pull drive.

Connecting switching regulators in series can be problematic because they have negative resistance (higher supply voltage leads to a lower current draw) it's best to connect both regulators to the battery.

if you're building this thing out of modules, use one of those class-D amplifier modules to drive the loudspeaker, you'll probably get acceptable performance operating it from 5v.