Do Achievements/ Trophies for games add to the experience of playing for you? Or do in-game challenges?

Achievements not so much for me. But for the most part I'll try to do in-game challenges. Borderlands 2 challenges added a lot of fun to the game for me. Same with the GRA challenges for Fallout: New Vegas.

No. Not unless they have additional rewards, or have immediate benefit.

For instance, I like badges in city of heroes, because getting enough of them could give you special powers that, while far from game breaking were still pretty nice. And you could select any one of them to display on your name to other players. Pokemon Black 2/ White 2 is much teh saem way; the achievements don't get you anything, but you can use one to show the other people you battle against (the one I use is Never Give Up "A Medal for those who don't know when to quit even when there's nothing they can do.")

In WoW I like them when they get me titles or pets or mounts or the like.

But things like Skyrim, and its steam trophies? yeah that's stupid. They don't do anything. They arent good for anything. You get them for doing stuff the game makes you do, or for arbitrary nonsense that is hardly trophy worthy. Sure, both Pokemon and WoW and City of Heroes would do the same thing... But they would also have some tangible, if minor, secondary function.

Steam trophies are just plain stupid.

Edit: Yeah, I've thought about this.

Some of them can add flavor, and even serve as a minor side quest after I've beaten the game itself. Stupid trophies and achievements like, "completed character creation", or "be killed by [lame enemy]" are just stupid though, and whoever implemented those should be ashamed of themselves. 70% of the achievements out there aren't even real "achievements". I don't care about them myself, but their petty state in today's video games bother me.