Broken rear axle, now held together by the quick-release? In the old days, taking the wheel out to see would have been the worst possible thing you could do if you needed it ride the bike. Release the QR and the loose ball bearings would spill all over the floor and putting them back was impossible. Maybe you can reconstruct a sealed bearing hub's axle after pulling the wheel but if I really needed the bike, I wouldn't gamble on it.

If I were you, I'd pick up another rear wheel and only then, pull out your current wheel and check the axle.

To follow DiabloScott's thought, maybe ride the bike, do that hard start and feel the clunk, glide to a stop, put the bike in a solid strand (or tie the front wheel to a tree) and try pulling the rear wheel back and see if you can get that second clunk. Look for any cracks between teh BB shell and the rear dropout, especially on the right chainstay. Special attention the the chainstay to BB joint and chainstay to dropout joint. If nothing there, I'd then look along the seatstays.

Ben