Breathing & buoyancy are only really relative when you're stationary in the water. When you're moving through the water (on the surface) there's little correlation.
On learning to swim underwater first, there's a problem in that you need confidence to try that, which usually only comes after you've learned to swim! So chicken and egg problem really.
Swim teachers don't teach underwater swimming for a very good reason; by hyper-ventilating, it's possible to fool your medulla oblongata into thinking you don't need oxygen yet, hence no need to breathe yet. There are anecdotal examples of people drowning through hyper-ventilating then not breathing in time to maintain consciousness. (Shallow water blackout)
You're right about the head down thingy, 'cos you have more of your head in the water, the rest of your body and especially your legs are higher in the water, requiring less downforce to maintain yourself on the surface and you are more streamlined, hence why competitive swimmers spend more time swimming with their faces in the water.
Putting your understanding of physics into practice;
You weigh a definite amount, which displaces a definite amount of water, the more of your body is out of the water the more gravity/fluid dynamics or whatever wants to find a more stable state. Hence the average person will find a natural floating position with a little bit of themselves actually sticking out of the water, perhaps the crown of their head, or if in the crouched/mushroom position, it will be a fraction of their back, or if lying on their back it will be just their face, etc.
Next time try treading water, then raise one arm, see what happens. It's pretty obvious, you sink further into the water becasue you are now displacing less volume, for the same weight.