Does Amplifier's Sound card exist?

No not that I know of. The only thing that a sound card does is pre-amp voltage. Not enough room for what you are talking about. That would take up the whole case just about.

There may be but I've never heard of anyone doing it. The big thing again with this is space and then heat. Having a dedicated amp housed with pc components would produce an ungodly amount of heat. You would need high CFM fans. They produce alot of unwanted noise. You may get away with a liquid cooling solution but again space would limit you with that. Most build an HTPC and connect to a dedicated Amplifier. I'll see if I can find anyone trying what you are wanting to do. Who knows? I'm sure there's someone who has rigged up something.

OK. I haven't found one person/company offering 5.1 amplifers combined with a pc within one case. What I did think of though may meet your need depending on your taste in high fidelity components. If you are looking for a decent sounding home theater system and your not planning on using it in a large space then I would recommend looking at amplified speaker systems. Here is a review of a 5.1 wireless(rear) THX certified system from Logitech that had great reviews.

http://www.pcmag.com/article2/0,2817,1904834,00.asp

Other than that I think that you are going to have to have a seperate dedicated amplifer to run you speakers.

When you get it done I would like to see the finished product. I too have thought about this and think that it could be done and be aesthetically pleasing.