Do all objects get larger in an "expanding universe"? I'm envisioning an analogy of the universe as a plane with marbles in that plane to represent different planets.

Now, if this plane expands how does this work? I assume that it is not like 'nothing' becomes 'something' then but rather expanding/stretching the universe; but doesn't this imply that everything around us is actually stretching/becoming bigger (like every atom/quarks) becomes a little bit bigger? Or is it just the space between those subatomic particles and combinations of these become bigger? 

An example could be the structure of carbon dioxide becoming larger, but only because there is more space between the carbon and the oxygen atoms.
It would, if there were no interactions.

However, everyday matter is held together by other kinds of forces (electromagnetic, mainly). These are quite strong, and any "stretching" of space will be counteracted by these forces "pulling" the atom/molecule/object back to its original size.

So the expansion of the universe only matters on the larger, cosmic scale, between objects like galaxies which are separated by a large enough distance that they don't affect each other too much, gravitationally speaking.