I managed to break a pint glass while washing it and the gash down the outside of my hand under my pinky was quite bad (flesh hanging down and copious blood) - I only mention this because at the ER they told me it had to be stitched up and they wanted to put local anaesthetic on it (which I know hurts and makes the area swell so would make stitching twice as long a process) so told the doc to just stitch it without anaesthetic - he balked at first, but then I told him to not be a wuss and do it. It was fine, but then I have a very high pain threshold - I have endometriosis. Afterwards the doc told me he had worked in South Africa and they saw a lot of gunshot/stabbing victims and most would not have been able to do what I did.
Maybe I am a bit of a curmudgeonly old lady 
he does sound like a pain in the arse, however.
Ouch!
As for the other comment about women taking pain better, I've read that estrogen is a pain killer. When I had a lot of estrogen, little things never hurt. Now though, with less estrogen, I can see why some men bawl over papercuts.
I agree with your relative about latrine dumps from planes. I have smelled urine as planes have flown over. Not very often, though.
Sometimes it's better not to get stitches. Stitches stop gaping wounds and prevent scarring. But they also do a lot of nerve damage in fingers, and can lead to neuromas. As long as he controlled the bleeding and gets the wound to stop gaping, he will probably be fine. I don't avoid hospitals but did you know THE NUMBER ONE CAUSE OF DEATH HERE IS HOSPITAL ERROR, OFFICIALLY! Just walking into a hospital, one risks getting sick or fatally sick. Hospitals don't allow daylight in, to kill germs, and they don't allow fresh air in, either. Many buildings don't, and society ends up with building-specific outbreaks of norovirus and Legionnaire's disease, which are potentially fatal, especially in the elderly and infants. I have met three people (2 men and one woman) who recently acquired antibiotic-resistant superbug infections that required deep cuts and disfiguration to remove from their flesh. They were fortunate to live. Hospitals know that hand sanitizer is vital in reducing infections. They know that the hand sanitizers which kill e-coli are not the same hand sanitizers that kill the flu. When it comes to hand sanitizers, and germs, there are at least two different categories. But instead of making employees use two sanitizers, they only use one, leaving the e-coli end of the loop open. THEY ALL, if they weren't so evil, could simply use a vinegar solution to kill e-coli type germs.
I didn't quite understand how your relative came to nearly cut his fingers off! But based on what you said, he might have obsessive-compulsive disorder (ocd). I would discreetly limit his access to the dog when he is blotto. Hopefully he wouldn't ever hurt the.dog, but it sounded like maybe he was resisting a compulsion.
As for being charged for the dispatch, what state are you in? Your state might have a law regarding "duty to call for aid", also known as "duty to rescue" laws which might have required you to call for help for a gaping wound. Also, your state might have a good samaritan law, which might possibly exempt you fro having to pay for trying to get him help. You also might have a case if he was drunk when he refused help, trying to help someone who was too incapacitated to make their own decision. It sounds like it would be worth it to at least write a dispute letter and see how far you get. Most lawyers will give a short, free consultation over the phone, even if they don't think you will hire them.
The interaction between him and the dog as he cut himself sounds like a mental illness. Document the date, time, place, and names and contact information every time something like this happens. Because if there is a pattern of him being a danger to himself or others, then mental health treatment can be forced upon him. He could be taken to a facility for 72 hours, and possibly medicated.
Also, if he ever becomes legally incompetent, documenting now will make any future incompetence proceedings easier. Some people have a guardian ad litem who makes decisions for them. Others have a conservator who controls their money. Read the National Association to Stop Guardianship Abuse, because the system can be okay, but also can be so crooked as to deeply hurt/abuse/rip off the citizens it is supposed to protect.
He sounds like an entertaining character, and a people person. He probably needs a support group of vets, treatment for any psychological disturbances, and to go easy on the alcohol. Whether or not a liter is a lot, he sounds like other alcoholics I've known. It would be kind of you to present foods he likes when he is drinking, to help his brain and organs. If he is just a character, though, there is sure nothing wrong with that. His speech is harsh, but hopefully he wont tick off the wrong person. Him being at home might be good for him.
