Joshua Aslan Smith makes a number of completely reasonable assumptions in his answer, but I think it's important note that they are in fact assumptions.
Any Teleport You Make
I'm not inclined to assume this means that you are the one that must teleport. That would be totally reasonable, but I can't say for sure that it's the case. I mean, if I teleport somebody across the room, then that was a teleport, and I made it.
Maybe Smith has a rules reference from which he drew his conclusion, but he didn't say what it is and I don't know about it.
I would discuss the interpretation of this sentence with my DM and make sure that everyone is on the same page before trying to use these boots.
What is a Teleport Power
Once again, it is completely reasonable to assume that Wizards intended the Gloves of Dimensional Repulsion to affect powers with the Teleportation keyword, but we don't actually know that for sure. This assumption would be consistent with the way in which most effects determine which powers they interact with: keywords. In this case, Smith is completely correct that Walk with the Fey does not add the Teleportation keyword to powers.
It would also be reasonable to point out that the Adventurer's Vault has a 9 page errata document available on Wizards' site, and nowhere in there will you find a word about these gloves. If they're going to take the time to add keywords to powers, you would think they would also make any changes to effect text that interacts with keywords. In this case, I would say a 'teleport' power is any power which teleports, which would be true of any slide power while wearing these gloves.
Either way, it's an assumption. So, once again, discuss this item with your DM to make sure you're on the same page about how it works.
I believe that they're referring to restrictions on the destination, since they specifically say "Where you teleport the creature remains subject...", but maybe not. Smith seems to think that this would include generalized restrictions, even going so far as to suggest that a creature would get a saving throw for being teleported across dangerous terrain.
If my understanding is correct, then at first blush the clause seems redundant. They already say "an equal number of squares", however the number of squares is not always the only restriction. Sometimes they have to end adjacent to you, as you pointed out, or something along those lines. The "up to K spaces" restriction you asked about is covered by the "equal number of squares" clause. If you can slide the target up to 5 squares, then you can teleport them 0, 1, 2, 3, 4, or 5 squares.
So, you could teleport through enemy spaces, through blocking terrain, past dangerous terrain, and so on, just so long as the final destination square meets any restrictions on the slide's destination square.