I thought the room at the Inn could be used to store items; but after progressing the storyline a bit the innkeeper decides its a new calendar month and I need to pay rent again. When I do this; my room is completely empty of items!

Is there anywhere items can be stored?
If not, any spoiler-free tips for what's worth holding on to (and I'll just sell the rest)

I would test it in a portal world. To my knowledge after you complete them the map itself does not change. Preference would be lava world.

Gladstone Woods, Gladstone (all areas) pretty much change with every world that you complete.