Can anyone explain to me what is carried over to a new game if I don't meet the shop's payment?  Do inventory, store items, cash, and merchant levels carry over?

From what I remember, pretty much everything but your cash is retained. So if you're like me and you fail at the very end, the restarted game becomes very easy,

Do you mean New Game+? 
 
Because you'll bring over unsold inventory... that you had on you... I think. It was hard to tell, but I totally had some high level inventory on day one. Any adventurer True Card comes with.
Your shop reverts to the original size, but you'll keep your wallpaper, floor and counters. As well if you had vending machines they'll be in your inventory. 
 
You'll have all your xp and levels, but none of the cash. The guild master will carry your current level inventory... but it seems like the highest level stuff is sold out for like weeks. Not a big deal though, you can just sell the shit out of Grand Naginatas. 
 
The game world sort of resets around you, you don't get news alerts and price adjustments until like day 7 or so. Which sucks because that's how I normally make money. 
As well you'll have all the story cut scenes to skip through.

Does anyone know how I can leave the dungeons without actually being killed? 

As far as I could tell it was just about everything except for cash. The last time I played I was up to my second payment of 30,000 but I only had 27,000 and no matter what I did I could not raise enough cash, so I looped and lost all my money. Thankfully I saved before that, reloaded and just bought as much shit as I possibly could since your inventory carries over. That way if you can't make a payment, you aren't at a complete lost and you can build up a sizable inventory for your next run.
 
@ch13696 said:

" Does anyone know how I can leave the dungeons without actually being killed?  "

" @BlaineBlaine: My shop didn't revert to the original size, so I think you keep that too. "

Yeah, I barely missed the last payment during my first playthough, but I kept my inventory for the restarted game. When I went through the game again, it was rather easy (I'm pretty sure I ended up with something like 4 million pix at the end of that playthrough).