Breathing properly is critical. Air is the fuel for the voice. If you don’t know what to do with your breath, nothing else will work.
The second question is interesting. Most people actually breathe just fine on their own without thinking about it. But when actually going to sing, they tend to mess it up, because they are thinking about it too much (and trying to take in too much breath). So it tends to require extensive training, in order to do what you already know how to do. Sometimes the brain is a bitch.
I don’t know specifically what happens physiologically, but it’s basically the same principle behind needing to warm up your muscles before working out, especially first thing in the morning. I suspect it’s a combination of deep relaxation and dehydration making the cords (which are, after all, muscles) thick and sort of floppy. Once the blood gets flowing, they’re a little more active, which allows you to get to the high notes, but also sort of cuts of the low notes.
A few weeks ago I woke up one Monday morning after a week of being ill and sang a low B-flat. Give that I’m now a tenor, and was never a bass to begin with, that was pretty fun. Never works when I need it, though.