Hi there.
Well you've managed to make me very confused
Why do you want to avoid using a good receiver? Especially considering it is the easiest, best and most effective way to do everything you say you want to do?
Your Asus Xonar DG sound card does not have a digital coaxial output, so far as I can tell. I'm just going by what it says online, but it appears to have 5.1 analogue line level outputs, plus an optical S/PDIF TOSLink output. That TOSLink optical output is your digital output from that particular sound card, which is perfectly fine. There's no coaxial digital audio output though.
Next, a simple DAC would not be enough to make sense of that optical digital output - unless you limit your output from your computer to 2-channel PCM only. You clearly want to use Dolby Digital and DTS surround sound, so this is not a good option. You need something with a Dolby Digital and DTS decoder so that you can take the optical output from your Asus Xonar DG sound card, send out the 5.1 Dolby Digital or DTS signal, and have that other device do the decoding and digital-to-analogue conversion so that the signal can then be sent to an amplifier, and then to your speakers. The easiest way to do all of this - by far, and especially within your budget - is to use a good receiver. A good receiver from Onkyo, Denon, Marantz, Yamaha or Pioneer can provide you with an optical digital audio input, the necessary Dolby Digital and DTS decoders, DACs that are every bit as good as any outboard "audiophile" DAC, the necessary 5 or 7 channels of amlification, plus the necessary processing - such as Dolby Pro Logic IIx - to turn the 5.1 signal coming from your Asus Xonar DG sound card into a 7.1 playback experience.
So, yeah, I feel as though you must have heard from someone or read somewhere that an A/V receiver is somehow "inferior" in some way. I've no idea where you would have heard that. There's nothing that a separate, outboard DAC can do any better than a good A/V receiver. And an outboard DAC CANNOT do everything that you need and want in this case. A good A/V receiver can. So that is clearly your best solution here.
If you are happy with the speakers that you have already, you can buy just a good A/V receiver. If you want to get all new speakers, you could opt for a HTiB, like you suggested, or get an A/V receiver plus a new speaker package.
Since you're in Canada (I am too
) there aren't quite as many good online deals as there are in the USA, but you can check FutureShop, Best Buy, Visions, electronicsforless.ca and HD.ca for current prices.
I tend to favor Denon, Marantz and Onkyo - mainly because I prefer Audyssey for auto-setup and EQ vs. Yamaha's YPAO or Pioneer's MCACC. But Yamaha and Pioneer have very good receiver offerings as well.
Anyways, that's your best solution. You need an optical input, Dolby Digital and DTS decoders, DACs, Dolby Pro Logic IIx processing (if you want to expand 5.1 to 7.1), a subwoofer output, and amplification for all 5 or 7 speakers. A good A/V receiver gives you all of that within your budget, plus HDMI switching if you want it. I've no idea why you wanted to avoid a receiver, but if that desire was based on something someone said or something you read somewhere, all I can say is that that was bad advice
Best of luck!