You would not have to take expansion into account.
Gravity is strong enough to totally blow away the effects of the expansion. An example of this is M31 and the Milky Way coming together. This is at odds with an expanding universe. It is an example of small scale anisotropies that exist in a large scale isotropic universe. Zoom far enough out, though, and the big picture shows what you'd expect. So yes, the balls can move freely on the sheet within reason.
And yes, eventually everything will be torn apart by the expansion. This theory is actually in vogue right now. Starting with galaxy clusters, then galaxies, then open clusters, then stellar systems, then planets, moons, asteroids, rocks, molecules, atoms, quarks....
It's not a pretty end
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"For thousands of years humans were oppressed - as some of us still are - by the notion that the universe is a marionette whose strings are pulled by a god or gods, unseen and inscrutable." - Carl Sagan
Freedom consonant with responsibility.
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