One of the names of this cactus is Opuntia subulata, I think - you'll often see a cristate version, too.

Nice cactus;). It's a Mammillaria what species, I have no clue. For me Mammillarias can take forever to id. There are so many species and then hybrids of those species. Blooms on Mammillarias are usually small. Yours has similarities to M. backebergiana:), but your columns are smaller.

I would start by getting it out of that cup. Get a pot with drainage holes(terra cotta preferably), cactus mix, and perlite or pumice. The new pot should be about one inch wider than the cup. A mix of 60% pumice or perlite with 40% cactus mix. When repotting get all the old mix off the roots. Don't water for a week after repotting. Now when it comes to the etiolated growth it would probably be best to cut those pieces off. Once you give more light the old skinny growth won't be able to support the new sturdier growth. Just take a clean knife and slice off the lanky growth. I would do this after you've repotted:). Do you live in a dry climate or humid?

I think it is a Mammillaria. The species I found that looked closest are: freudenbergeri, lloydii, roseoalba (but flowers light pink), uncinata, and winterae. M. uncinata had the most similar characteristics to you picture.