1. Does it really need EAX? My onboard soundchip doesn`t support it. Afair from the boxed version it doesn`t need EAX, it`s optional, right?
2. Does it work correctly under Win 7? I heard that the game has problems under Win 7 when it`s patched.

I'm running Windows 7 64-bit and FEAR runs really well on it. I don't recall any EAX issues, but EAX shouldn't be needed for the game.

I'm running Windows 7 64-bit too and it runs flawlessly. As for the sound card i use, it's an external Focusrite Scarlett and i don't have any problems with the sound. But if i remember correctly, years ago with my old PC, i had some sound problems using a Soundblaster but i don't remember exactly what was about and if/ how i solved them. (maybe it was another game too!)

If you run into sound problems, make sure you have your soundcard driver updated. ;-)

I have the Steam version so if GOG's version is completely different it doesn't apply but I doubt that. No it doesn't need EAX but you only have stereo on Windows Vista and higher because of changes made in DX (read Microsoft's PC degrading strategy in favour of pushing their console).
If you want surround under Windows Vista and higher but you don't have a soundcard from Creative, you might want to try Indirect Sound which gives you surround sound but still no EAX. The game should work, I have read on several sources that the game does have framerate problems with certain keyboards and mice of Logitech's G-series, but I haven't encountered them myself as I don't have that hardware.

Regarding Windows 7 and Logitech...

I just started playing on a Windows 7 x64 laptop; it is working fine so far, outside of the Logitech issue. I do not have a Logitech G series, but the problem may be in the Setpoint* software. The workaround for the issue is disabling the unspecified "HID-compliant device" listings in the Device Manager. That disabled the custom mapping I have for a mouse button in Setpoint but seemed to fix the low FPS issues.

For example, I was getting 13 fps with the HID devices enabled and 60 with them disabled...

( * I know it isn't called Setpoint any more but I can never remember the new name.)