So I just chuck everything remotely recyclable into it, plastic bags, paper bags, junk mail, food containers like yogurt cups and soup cans, packaging, wine bottles, whatever. How do they sort that stuff and do you need to wash stuff out still? Like soup cans, don't they wash them? My mother in law still washes stuff and removes all the labels. And something like a pizza box that's soggy with pizza grease. Is that still recyclable? And if not and you put it in there does it contaminate everything else so it's all wasted?
That is dumped into a truck, which dumps it into a 'first stage sorter.' The first stage sorter is a bizarre contraption with a bunch of belts that haul the stuff through an array of stations where various forces are applied that will have different effects on different material; magnetic, air flows, vibrations, all kinds of stuff. Cans jump, papers fly, the beltways split, and split again, and again. And sorted stuff is delivered to the input stages of the various processing equipment.
So to answer specific questions you have to look at two things; will it interfere with the sorter, and will it screw up the actual processor.
Your soup can should be washed out, because if it has residual soup gunk dried up in it that is extra weight. In the sorter magnetic forces are used to knock steel cans out of the stream, but the force used has to be carefully balanced so that it knocks out the cans reliably but doesn't knock them too far. Extra weight that isn't steel can make the sorter miss a can. Removing the label isn't necessary because the sorter is set up to accommodate that negligible difference. The processor will burn off any labels. It will also burn off any food residue, but only if the can makes it through the sorter properly.
Your yogurt cups and such will get blown off the main belt at some point, and if they are weighted down with dried food that can affect them too. They don't have to be clean, just don't leave them like half full of extra weight.
Glass bottles and jars are vibrated out of the stream because glass is the heaviest thing left after the steel is removed. Since glass is sorted out through processes that use its weight, food residue in a glass jar actually doesn't reduce the effectiveness of the sorter. Your jars and wine bottles should be uncapped so the lids can go where they need to go rather than being dragged into the glass processor. The first stage of the glass processor breaks up the glass and then it passes through a secondary sorter that will pull out the metal lids, but the lids will sometimes cause problems in the crusher that breaks the glass, and the secondary sorter at some recycling plants doesn't feed the lids back into the steel stream very efficiently so be helpful. Again, the glass processor will burn off any labels and food residue so don't stress about peeling them off and cleaning out that last bit of cheese dip in the bottom of the jar.
The number one cause of shutdowns in sorters, by far, is thin film bags that get caught in the works. Those really cheap, really light bags that are generally used to bag your groceries fly around in the lightest breeze. There's no effective way to get them out first, and there's no effective way to keep them from drifting along with whatever is being removed from the stream at every step, so they wind up everywhere. They get jammed in gears that drive the belts, fly out of the recycling center on the breeze and infuriate the neighbors, jump in and out of various high temperature processors and turn into flying incendiary weapons...no kidding, they are the reason that recycling centers and programs get abandoned. Do not put thin film bags in unsorted recycling.
Then there is the pizza box. It will get sorted just fine in the first stage sorter and come out with the rest of the cardboard and paper. Now, take two pieces of paper, get one soaking wet, and then cut them both with scissors. Notice how the wet paper doesn't cut, so much as it just rips into a soggy mess and gets stuck between the blades of the scissors? That's what the pizza box does in the shredder. Cleaning gooey pizza box residue out of the shredder blades requires shutting down the paper processor after some limited number of pizza boxes, making them the second biggest drain on the efficiency of your recycling center. Compost them.