In the early days of HDMI implementation, a pass-through from the sound card to the video card was necessary if you wanted a PC to output it's audio over HDMI. However, having external devices source content for playback of protected content, such as Blu-ray movies is a no-no, so the hardware source for protected audio is now built into graphics devices, entirely bypassing any other sound equipment you may have in the computer. The sound card simply isn't used.
Since your computer has both a dedicated sound source for audio sent via HDMI to your receiver, and also a sound card, you actually have two separate sound devices that Windows has to choose from. Plugging headphones into your sound card is not going to duplicate the sound coming from your HDMI just as the HDMI will not be reproducing the sounds coming out of any headphone port you may be plugged into. If possible, I would recommend sourcing your headphone from your receiver.
Your soundcard is going to be performing the digital to analog conversion for any headphones plugged into an analog output, so yes, the quality of sound card will directly affect the final quality of your results. Whether people will hear the difference between expensive equipment and on-board is very subjective, and may rely as much on the cans being used as the sound card, and I suspect that even if on-board was capable of perfect audio reproduction, sound cards would be considered superior due to their coloring of sound and peoples' bias.