Shorter fibers pass through the screen and do not get incorporated into new paper. The longer, stronger fibers, DO get incorporated into new paper. So, the average fiber makes it through the system eight times. No, the ream wrappers for copy paper has a thin layer of plastic film between the outer layers of paper, which makes them unrecyclable. Paper plates, cups and napkins cannot be recycled because of contact with food waste but also because the paper fibers are too soft to be recycled again.
Disposable dishware is just that, disposable.
The machines that are used to screen the metal caps from soda bottles can also screen the metal rings out of tennis ball cans. Plastic caps are made from a different type of plastic than the jugs or bottles and cannot be recycled together because they melt at a different temperature and can ruin a batch of plastics being processed for recycling. Other plastics are not acceptable. During manufacturing, air is literally blown into the container to inflate them like a balloon. They cool very rapidly.