those are very kind a good replies. Many thanks especially to lynwill.
at the end, I would love to be able to power this with a single, standard 18650 Li-battery, which can provide a voltage down to 3.6V.
Based on the loudspeaker's spec, I assumed just 0,1Volts (RMS) would be already sufficient (the spec says 106dB for a 1kHz tone would be achieved with 0.26mW. Together with the impedance of 50Ohm, I believe according to sqrt(P*R)=sqrt(0.26 mW*50 Ohm)=0.1V is fine.
Regarding bridged configuration: after I read several TI's spec, I got the impression "bridged configuration" is the right way to go ... those datasheets are only mentioning the advantages, but not the disadvantage. To be honest: I have to do more "google-research", to understand when to use which configuration (also to understand what it means "the common" has a live signal"). Again, I just chose bridged, because the TI-datasheets made me believe this is the best but I will try to catch up and do my homework.
Regarding vol. control: based on your description (which I highly appreciate), I will try to update my sketch, so others may also be able to benifit from this sketch.
Regarding coupling-capicitor: I also have to think through this, but also here: thanks for the hints (and give me a bit time to understand this).