If you haven't already, I'd recommend you check out this topic, there's some discussions and comments from DMs as well regarding the fey:
http://bgtscc.net/viewtopic.php?f=49&t=30656
More specifically to answer your questions:
- What benefit do the archfey get from (a piece of) your soul?
I'll just start out with saying, I don't know for sure. I'd imagine that they use it along the same lines as the Nine Hells and the Abyss does. For power. They can turn it into a sort of fuel to increase their own power.
Unlike the Abyssal and Infernal pact however, I doubt you get a chance to become one of them... so you're likely worse off than if in the hands of devils and demons

That's just my own private thoughts, if someone has some actual lore to substantiate their claims you should listen to them.
- Do we have fey pacts in BGTSCC?
Yes, we do.
- A bit more generic: Can someone explain in short, why someone would be bothered by not owning his/her soul anymore?
Losing your soul, means losing your afterlife. If your character is faithless or doesn't expect anything good to happen to them in their afterlife, it may not seem like a big a deal... Except that it makes it more difficult to change their mind later, but that's your characters future dilemma, in exchange for power now
I hope that answered at least some of your questions!