I picked this up on PS3 for a couple of bucks on sale (which is still current at the time of this posting by the way) and having a ton of fun. I hit a part which is a bit harder and I noticed earlier the game saves inventory separately. The question is: can I, after selecting my save, pick a new game option and start over with my inventory and upgrades?

Yes you can, due to the way the inventory/NG+ elements work, anything you pick up stays with you on any playthrough, so you could be halfway through the game on your very first play and go back to any chapter you've unlocked (on any of the difficulties) and you would still have what you acquired that you haven't used/gotten rid of.

@zolroyce: Great, thanks!!

EDIT: Just to confirm for no good reason, I started over and everything carried over. The part I hit before was not even that hard, but I was in a mood to just put a podcast on and grind some upgrade money. And with the chapter select feature this gets even better and more flexibe, not having to straight up restart.

All and all this is a pretty great game, especially for 6 bucks!