Yes, you can only display designs in your town that are currently in the 10 active slots on one of your characters.
You can save designs via displaying them in the Able Sister's Shop and/or by talking to Mabel and asking her to save a design for you. However, if you move Design 1 in Slot 1 to Mabel's save, then put a Design 2 in Slot 1, Design 2 will be the one that shows up in the game. Design 1 will still be accessible, but will show up in-game only if you go back to Mabel and place it back in one of your active slots.
In order to get around this limitation, you can create other residents in your town and use them essentially as "design mules". Making a total of 3 other residents will give you another 30 active slots to play around with.
Otherwise, yes, you are correct, the other rooms can be essentially used as museum versions of your house (which won't be graded by the HHA, obviously). Given that you get special carpets for turning in X amount of Bugs, Fish, and Fossils, I'm personally going to be using those rooms to display extras of those things that I think look neat. I'm also making a Nintendo room. It's mostly extra space to give you opportunities to show off your collections.
Best Answer
As this page explains, to unlock the second floor you must have "talked to Blathers on at least 14 different days, donated at least 20 items to the museum, with at least one in each section (insects, fish, paintings, and fossils.) Once the criteria has been met, Blathers will be seen in a thinking pose. Talk to him to unlock the Museum Renovation public works project"
Once you complete that public works project you will get the second floor.