Hi DC, In answer to your questions:-
1, Darkbrowneggs has explained checking the pelvis width. The red comb ones should be laying. The big poo ones should be laying (big poo first thing in the morning ones). Trap nesting is the only certain way -hen is trapped inside and you mark it and release. Or you watch them, catch them and mark them.
2, We cured all our egg eaters with a pot egg. They bash their little beak until they have a headache and give it up as a bad job. Filling an egg with horrid stuff doesn't work -they love mustard and the like. The only hen we failed with used to lay and eat her own egg immediately. Haven't tried our rubber eggs yet. If its a rat or Magpie it still works
3,Mucky eggs can be washed as long as the water is warmer than the egg, colder than the egg and you wash disease into it. Problem is you wash the film off as well which makes the egg porous and therefore it dries out quickly, so eat within a week, rather than the month. The film is created by the lubricant the hen produces to ease the laying, which then evaporates leaving a film covering. Root causes in your case are not removing the eggs soon enough before another hen with dirty feet goes in and letting them roost in the nest boxes, fouling them up. The latter is a habit that needs breaking, both because of dirty eggs and encouraging red mite/ lice in the nest boxes. Block the nest boxes off at night or pick them up onto the perch in the dark.
You are not allowed to sell washed eggs anyway, you can only scratch the worst poo off. If they are for resale they need date marking as well. The regs have changed recently so worth checking you comply.