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Because his penis had been out as long as it has, he officially has a problem called paraphimosis. This is an inability to withdraw the penis back inside of the sheath (the prepuce). This condition if untreated may lead to severe penile trauma. The penis rapidly becomes dry, cyanotic (blood flow to it is cut off), edematous (swollen) and painful. The penis can become infected causing a system wide infection called sepsis. It can also become necrotic (dead) and actually fall off your dog after some time.
I own a Maltipoo that had the exact same problem with his previous owner (she relinquished ownership of him after she didn't want to pay for the surgery to amputate his penis and save his life) who waited for 2 days before seeking medical attention. He ended up needing emergency surgery called a perineal urethrostomy...essentially this means they amputated his penis and testicles and re-routed his plumbing giving him a little urination hole out the back.
Sadly, you have already tried all the at-home suggestions I would have offered you. At this point, he is absolutely going to need to be seen by an emergency vet immediately. They'll get him on some antibiotics and pain medications and will likely sedate him so they can try something else to get his penis back in the sheath.
I'm sorry to have to deliver such bad news.
No, there truly is no home treatment..,.and icing it can actually cause a bit of frostbite, so let's not use that.
If this had only been going on for an hour or two, I'd tell you to go ahead and just monitor it and see if it improves, but since his penis has essentially been without any blood flow for 24 hours, there is a very real possibility that this could require surgery to repair. The longer you wait, the less likely it is that your dog is going to be able to keep his penis.
I think you should let the vet take a look before we even remotely worry about surgery.
However, as long as he's in generally good health, then he should do fine in surgery and recover normally.
I would not let him lick...although I understand why he's doing it. The problem he's having right now is very, very painful (think about times you wrapped a rubber band around your fingers and watched them turn colors...now imagine that on a penis), so getting him in to the emergency vet is the absolute best option.
I am so happy to hear that Bear is doing better and I commend your decision to get him neutered!!
Give him a pat on the head for me!!