Aww, poor fishy
1; It's hard to tell how painful dropsy is; it's more likely to be uncomforatable than painful, but that's only my best guess...I could be wrong.
2; Dropsy is not usually contagious,; it's more a symptom of something wrong rather than an atual disease in itself. Have you checked your water quality?
3; Hmmm...well, once the fish has started 'pineconing' they don't usually recover, I'm afraid

You could try Epsom salt bath(you do need the proper Epsom salts; aquarium salt doesn't do the same thing; you can get it from a chemist) 30 minute baths in a tablespoon of epsom salts dissolved in a gallon of water.
There are a few methods of euthanasia. Please don't flush it or put it in the freezer; these methods are probably painful for the fish

If you're braver than me, you can give it a good whack with a brick; fast and painless as long as you crush the skull
Or, if you can get some clove oil from the chemist; put the fish in a litre of tank water. In a small bottle, mix at least 25 or 30 drops of clove oil with a little of the water and shake until it forms an emulsion.
Then you just add it to the fishes container; the fish will become unconscious within 20 seconds and will be dead within a minute. I'd just leave it there for half an hour or so, just to make 100% sure.
Obviously if the fish is bigger, you need to multiply the amounts, but 25/30 drops per litre is the lethal dose.
Last time I posted about this, someone recommended putting the fish in the freezer
once it was already unconscious from the clove oil and that's probably not a bad idea