So, to be clear, Dinosaw, do we have two boys, one all white (front of the perch in first pic) and one with a creamy collar behind the hen in the third pic.?
-and the 4 girls are the all- black one and the all-white one, ( both behind the white cockerel in the first pic,)
plus the Sussex with the black necklace in the third pic, in front of the cockerel, and the cream and white one in the foreground of the 4th pic?
I agree the scaley leg is really bad on all of the pics, and this in itself is a sign of age - it takes some time to develop as far as that. It can be treated, but it takes months of repeated treatments to eradicate it as the mites get deep under the scales on the legs and lay their eggs, which keep hatching out and re-infesting after the adults have been treated. If I were taking them on, I would want to keep them separate from any new birds I got, because the new ones would catch them from the infested ones when roosting together in the same coop. And also they may well have other parasites such as chicken lice (which will show up as white eggs round their vents, Rick recently posted a picture of this) So, if you seriously want to keep chickens properly, I do agree with you that one way to go would be to start with healthy, young stock if your own choosing, in a lovely clean mite-free coop.
This gives the problem of what you do with the ones you've got. An experienced keeper might cull them painlessly for you, if you know somebody who could help, but it's not a good idea for someone to try who hadn't been properly instructed, and it will be more difficult to tackle the cockerels as they're larger birds. My vet would charge me £15 per bird to put each to sleep, which would be a peaceful end and this would include disposal of the bodies. There might be a reduction for six, and anyway the prices would vary locally, or a vet might be able to point you to a local farmer who could help you out. Not all vets are good with chickens and finding one may not be very easy.
I just keep a few pet birds for pleasure and for eggs, and I let them live out their lives in retirement, and once they're too old to lay, I have them PTS once they get anything terminally wrong with them. But that's after they've been carefully looked after by me as pets for several years, - in your position, with these neglected birds who are not your responsibility, I would probably feel it was good to have a clean sweep and then start afresh. I would definitely cull the cockerels, and agree with Dinosaw about the black hen. It's a shame, really, they're all quite pretty birds. They've had quite a nice life, though, haven't they, not ideal, but plenty of room to roam and enjoy the sunshine for a few years, unlike battery hens, so no need for you to feel bad about giving them an easy way out. I might send the bill to the person who left you with the problems, though!
And definitely BURN that horrible old coop, when the new one is ready, preferably ON the spot without moving it, as it will be full of red mite and their eggs, the only way to kill them is by fire, and if you attempt to dismantle it and move it, you'll shake them off on to the run and they'll just crawl back up again on to the nearest chicken....