If I'm making a large house mod and have all rooms of the house in one cell, with lots of objects, will this cause problems?

For example will objects that I place around the house, let's say I fully decorate the house and fill all the empty space/cupboards/display cases with weapons/armor/misc items, stay put or move on their own and get messed up?

Havoc wreaks havoc with everything, even over your best efforts to prevent it. Best practice is to not place clutter too close together, or too much clutter in small spaces. For example, don't have books touching each other in bookshelves or if you want a pile of plates on a shelf, leave a tiny space between them. If you're not sure if you're over-cluttering there's the Run Havoc Sim in the CS that can test the effect for you.

As for your first question, it depends on the house. I learned the hard way that trying to put the Leyawiin Fighters Guild rooms all in one cell was very bad for performance, which is why that building has load doors in its interior - so that those rooms were in their own cells. Try to keep your clutter to a minimal level, and don't over-use light objects, those are the worst performance busters.