I'd really suggest confirming what the problem IS. That would be the first place to invest your time and energy. Just a logical guess isn't the best basis on which to proceed.
Misting sevin on the outside of the chickens' plumage won't do much. If you have mites or lice, they mainly are down at skin level (or in the case of roost mites, they are on coop furnishings rather than on the chickens during the daytime).
I would absolutely totally NOT dip or drench them with liquid sevin unless you happen to obtain a preparation FOR POULTRY that SAYS SPECIFICALLY to use it that way. Sevin is not the least toxic of all things in the world, and just cuz it is relatively ok to dust their skin with *dry* sevin does not in any way mean it would be equally safe to put *liquid* sevin on them (even if it were the same concentration).
Really, use tested products as per label directions, don't "freelance". Unless you consider your chickens disposable.
As for decontaminating the coop, your suggestion of spraying sevin all around is unlikely to do very much good. Mites or lice, if any there be, are going to be HIDING, and unless you get them WHERE THEY ARE, you will just be wasting money and time, and exposing yourself and your flock to not entirely harmless chemicals for no good reason.
If you are going to treat the coop, first remove and dispose of all bedding, including nestbox filler and aisle sweepings. Then clean well (sweeping, vacuuming, scraping, damp cloth, whatever your personal style is). THEN apply your sevin to the BARE SURFACES, making sure to get underneath the roost and roost brackets and all around and in and under every part of the nextboxes and other furnishings. Then put in new bedding. For the run, remove all loose material -- although this is unlikely to be a big issue unless you have a serious infestation of northern fowl mites or lice.
If you don't do all that cleaning BEFORE treating, you are just sevin-ing the dusty shavings-y surfaces of things, not the bugs themselves.
Good luck, have fun,
Pat