That's got one of the finest common names in Cactusland, and I'm not a fan of common names, but 'Dogtail Cactus' is one I refuse to eschew.I think it's Deamia (Selenicereus) testudo, and it's a tropical cactus which has one of the largest flowers in the Family. If I'm wrong about the specific name, it's a Selenicereus for sure.

I wouldn't remove the pup now - wait a year or two, as it may help it flower sooner and it's not the right time of year to be removing it to grow. I'd repot it in porous soil - this in one of the plants which it generally is OK to use commercial cactus mix with, as it's more accommodating of peat-based soils, being tropical. Yes, repot it and wait a few days and give it a little water. This is the slow time of the year, so water it about once a month and keep it warmish and fairly well-lit (as it's junglish in origin, it's a plant that doesn't need direct light, just lots (preferably) of indirect light). When it's warmer it prefers outside, and you can generally water it every week or so when it's spring/summer/early fall.

Personally, I'd leave the pup on next spring and cut that long stem in sections and root them - the plant would benefit from removing that irregular growth, and you'd wind up with lots of this great plant.

Not Dogtail. If you weren't such a cat person McHarris, you would know that plant looks nothing like a canine tail unless it has had the mange for a while. I think the plant could be Selenicereus innesii, or maybe a Harrisia. I have a similar plant. I bought the seeds to honor my departed from US friend, J. Harris. Seems to me they are now lumped with the Eriocereus. E. bonplandi and E. jusberti look similar.

Guilty, but as many know here, and in my defence, Your Honour, I am a FOD (friend of dogs) too. In fact, the cats allow us a dog here. I did have my moments of doubt, and with all that written, amended it to '....I think...'. Thank goodness I got the sun part right, eh?

And that part about the seeds - warm and fuzzy begins to describe it.