We've got four greyhounds, and are experienced with fostering/adoption. All our current four are older and we lost two last year 
who were in their teens. A few points.
First, and most importantly, I'd want to go back to your vet for a discussion as several of the points you've raised can be addressed and improved. You don't mention if the continence issue is urine or faecal - I'm guessing urine from what you are saying? There are a number of medications that can help with this - you need to discuss with vet, and it may be that you need to try them out. They may not completely cure it, but can certainly help. You can also get special mats for putting on sofas,chairs, car seats, in dog baskets, acting as dog beds that are designed for incontinent dogs as they have a wicking middle so maintain dryness for the dog but also protect furniture and surfaces, and are easily washable. Lots on Ebay/Amazon etc. Also, make life easier for yourself and get a steam mop!! Makes cleaning more hygienic too.
Bin raiding is absolutely normal behaviour for any sighthound of any age, but combined with what you are saying about his diet, always being hungry and not keeping on weight, I'd seriously consider changing it. Break his meals down into smaller ones, so he has three or four smaller ones a day, and make sure he is getting plenty of extras like pilchards, chicken wings (raw), scrambled eggs, poached chicken, etc. Do you know exactly what he's sensitive to? I'd think about moving him onto a different food personally. Sounds obvious i know, but is he up to date with worming?
Back end going - what does the vet say? We had a darling boy who got a bit wobbly in the back end before he went - he struggled at times with stairs, but still enjoyed running, walking. If he has arthritis, is he being treated for it? Have you considered hydrotherapy? What's he like out on walks/runs?
Cloudy eyes - yes the short distance vision does go - one of ours has got this, and our old boy who we lost last year too. All you can do is make sure house is well lit, kept tidy so things are where they expect them to be, and no trip hazards. Dogs manage surprisingly well with restricted vision - again, stairs are tricky and getting in and out of car - you may need to provide extra help. If htis is recent though I'd want vet to have a look and seek advice on looking out for any worrying signs.
Doggy smell - this is odd. Most greys have such light, non oily coats that they rarely smell. How often do you wash his bedding/beds/coats etc? I aim to do all ours on a VERY HOT wash at least once a month, but in practice in the winter it is often more. Do you groom him? My bet is that the smell is either an untreated skin infection (maybe a yeast?) or is also because of his diet. Again, I'd discuss with vet, and consider having skin scrapes. Also how are his anal glands?
Teeth/dental - greyhounds are notorious for having poor teeth. Again, back to your vet. I'd discuss having another dental and see if any need removing. Untreated infections in the mouth with stinky breath could be a major reason for him not putting on condition, as it is too painful for him to eat, and can lead to infections elsewhere in the body such as the heart and kidneys. It is no kindness to leave this untreated. Our ten year old bitch who came to us as a foster last year had all but four of her teeth removed - the difference in her health has been dramatic. Her infection had got so bad that her throat and lymph glands were affected, and she was literally starving to death as she couldn't eat the food she was being presented with in a kennel. 
Modern anaesthetics can be quite safe for older dogs (subject to usual health checks) and your vet wouldn't recommend unless they were happy to go ahead. It might be for instance that if you were having a GA for dental work/extractions, the vet might recommend an Xray to have a look at back end, for instance. Always worth discussing.
Your vet should be doing an annual check up with your boy and I'd consider also moving it to six monthly (what we did) especially if multiple issues, that you need to manage. Your vet is the partner in all of this - if you own senior dogs, this is such an important relationship. I'd also be in discussions with them about how you want to proceed when the time does come for the last appointment. For instance, you may wish to PTS at home - much less stressful, surrounded by those your dog loves. We did this for one of ours last year, and it made a hellish time a bit easier.
Good luck. 