How can the
sand at the bottom of the sand bed remain alive?

Gravity causes nutrients to fall to the bottom, where organisms utilize them.
You necessarily wont have a lot of 'flow' going past the sand, but it is being
mixed by the organisms in it. I have a fighting conch that burrows into the
sand for this purpose. I'd highly recommend one, their neat little fellas.

If not, then why doesn't just ordinary non-live sand become
live sand naturally by just sitting at the bottom of a tank?

It does, just not as fast and with less variety of organisms. Buy the sand
bagged, then get a cup or two of live sand to 'seed' your 'dead' sand from your
LFS or reef buddy. Dump it in one corner so it can breed. This will increase
your variety of life.

Another question I have is (and I know this probably sounds naive
but)if I buy live rock from the store, from a tank that looks clean,
does it absolutely need to be cleaned before putting into my tank. If
it looks clean to begin with, what exactly do I clean off of it?

I'm new to this too. I didnt clean mine. I've read that you should smell the
rock and look for decaying matter, scrub that off. You dont want the rock too
clean because like the live sand, theres organisms living in and on it.

Thirdly, if I put live sand in a completely empty tank (well with only
water in it) how can I assure that the sand will remain alive before I
put anything else in it (whether it's live rock or fish)? For that
matter how can I be assured the sand is indeed live when I purchase
it?

The sand would stay alive as long as your parameters dont go out of whack, i.e.
temp. salinty etc. One of my LFS's keep's their imported live sand in an unlit
tank with just about 4" of water above it. I noticed after adding live sand to
my prexisting dead sand that on the edges of glass you will see burrows that
have been dug by worms. I guess you could tell its live that way, otherwise
everything is just so micro, it'd be hard to quantify that.
~John