Yes all weapons, ammo, armor, and extras carry over to new game +. This is not true for hardcore mode. Note that even in New Game + mode, there are new armors that can only be found in a second play through.
yes it does, as soon as you reach the store you can reclaim your stuff. the only time you cant transfer your inventory if you play hardcore (No new game+, no checkpoints, only 3 saves through the game).