That's weird, it should automatically save your score once you get an end screen (either dying or retiring manually in the "Employee of the Month" tab).
The corpse you retrieve being on top of the monsters is indeed a problem, although you can still attack them. I think the overall enemy movement has some room for improvement.
And yes, whenever you exit the dungeon, any item in your inventory that's not equipped is automatically sold and can be used to buy items in the "Marketplace". It also does go into your final score, as total gold. The total score comes from a variety of sources, like exploring more floors, getting more items and having more gold. You can see them listed separately at the end screen.