For instance, I do not have any example of recording made with a Behringer mixer…
Having said, the Alesis Multimix doesn’t have a built-in compressor like the Behringer mixer. Is this a big concern, especially for recording voices or an electric guitar with a mic before the amp ?
No, not a big issue. If you don’t have a compressor you need to be a bit more careful with your recording levels to ensure that clipping does not occur.
Compression can be a quick and easy way to avoid clipping, but set wrong it can create unfixable damage to the recording. Built-in compression tends to be more useful in a live situation for handling the unpredictable levels that frequently occur in live performances (though you still need to be careful not to overdo the compression).
Note that he is using a reasonable condenser microphone (the choice and quality of microphone has a big effect on the overall sound quality. Use a microphone that is appropriate for the job. Large diaphragm condenser microphones are good for flute, but for mic’ing up a guitar amp I’d prefer to use something like a Shure SM57.
Note also that microphone placement is important. He is close to the microphone so as to get a good strong signal, but blowing to the side of the mic to avoid “wind blast” on the microphone.
Also, see the heavy curtain behind him. That is to cut down sound reflections from the back wall. The acoustic properties of the recording environment make a huge difference to the recording quality - a recoding made in your kitchen will probably sound like it was made in a kitchen and not in a studio.
The natural urge is to blow big bux on a microphone instead of soundproofing the room. Big mistake. The room pictured is using a cheap Shure SM-58 microphone.