Integrated audio cards have come a long way and it wouldn't surprise me even if they didn't suck all too much. That doesn't come with the fact that every integrated one is good, though.
Regardless, you shouldn't be looking at 7.1 audio systems with that kind of money.
The best active speakers go for around ~200€, are 2.1. Anything pricier than that needs an amp.
For headphones... well, just read this