What society first used the bow-and-arrow? And did the technology spread or was it largely discovered and developed independently across the globe? Lastly, were there any major civilizations that did not employ the bow?

We can't really say for certain where it originated from, largely due to archaeological constraints -- bows and arrows are made of biodegradable material, so the best we can do is look for arrow heads (which could just as easily be spear heads) or preserved wood from swamps and such. We've found arrow/spear heads in Africa from 50,000 years ago, so that gives us a bit of a hint. We've also found tools that appear to be ~10,000 year old arrow straighteners in the middle east, but that one's up for debate too.

The oldest actual bows that we've found were from a swamp in Denmark, dated at about 8,000 years ago. Needless to say, that doesn't mean those were the first bows created -- those are just the oldest preserved bows we've found. When considering this finding, note that Europe has been scoured for archaeological evidence considerably more than most other parts of the world.

Most of the big bad ancient empires used archers, including Egypt, Persia, India, and China. Greece and Rome definitely had archers (the Greek God Apollo was the God of Archery), although usually they were from foreign or mercenary entities (such as Crete, as mentioned by dontfearme22).