1) no.

2) judging from the subject matter of this post, I'd say amphibian.

3) uh... no.

4) couldn't tell you.

1. Have you seen a salamander?

Yes I have! More than one, even. They used to be out in the yard hiding under rocks and things. They probably still are, but I've stopped overturning rocks to see what's underneath them.

2. Do you know what they are? (They're not reptiles.)

I do and I did.

3. Do you know anyone (not me) with a salamander?

As a pet, no. I don't really know that many people, though, and I don't ask the ones I do know if they have any pets at all. I do know someone with a pet frog or two, so he might have a salamander as well for all I'm aware of.

4. Have you ever known a vet to accept amphibians?

Nope, though I also haven't looked into it. Vets around here mostly deal with cats and dogs and maybe birds. I wouldn't have much faith in any random vet knowing how to treat an amphibian anymore than, say, a fish or a tarantula or a snake. Maybe they would know how, though?

Also, yours is way larger than the ones I've seen around here, and they also have different colours and patterning. Also, also, salamanders have been considered 'fiery' creatures, this is supposedly because they were often in brush which was gathered to make fires. When the wood was lit, the salamanders would flee from the heat and people thought that they had somehow been created by the fire.

Answers, in order!

1.) Regrettably, I must say no. They do sound like delightful critters.
2.) I believe I am generally unaware of their classification. In short, no.
3.) I have known of absolutely nobody that keeps them as pets. Which is depressing, because non-furred critters, I am told, make good pets and conversation pieces alike.
4.) Around in my area? No.

But I have seen that there are veterinarians who specialize in...non-standard pets, I'd suppose some people would say. Most of them specifically train to care for animals that are not birds or mammals, such as iguanas, tarantulas and the like. They're considered exotics specialists. You may wish to look into exotic specialists, if you haven't already, and if there are any in your area (from our talks, your area sounds built-up enough that there should be a few lurking around). Not a whole lot of vet offices tend to keep them on staff, since people who keep exotics are sometimes a hit-or-miss proposition. If you can find a vet that specializes in herpetology, you may be in luck. (Also, they may help you if you ever decide to keep tarantulas around; most herpetologists tend to work closely with arachnids and scorpions. My mom has now had a total of three tarantulas in her life, two orange-kneed and one Mexican red-kneed.)

I apologize for the wall of text, but if it helps you find somebody to take care of your little buddy, then my job here is done.