No. The graphics card has no effect on sound being transferred through HDMI
Yes a better motherboard will have better onboard sound. But you'd still see greater gains from a separate dedicated device.
There is technically no sound coming from your motherboard if you are running audio out through your GPU via HDMI or DP. There is technically no sound coming from your graphics card either, as no processing is taking place. HDMI is a digital transport, as is DP. It sends 0's and 1's and any processing is done downstream via the device receiving the signal. So if you are piping sound from GPU to TV, it's the DSP and DAC in the TV that will determine the sound quality you get, and not anything inside your computer. The only time your computer is actually producing sound is when you have an analog device hooked up to your soundard or onboard audio via a 3.5mm jack (or RCA if your device supports it, which some do).
We've been in the digital age for a long time but it still confuses people for some reason.
Cheers.
No problem. BTW if you are using a soundbar, receiver, etc hooked up to your TV, the TV isn't doing any processing either, as digital transports such as HDMI or SPDIF are used to pipe audio to these devices. With digital audio it's always the device at the end of the chain that processes the signal.
However there is something to be said for proper setups. For example, not all digital outs are created equal. USB is a better audio transport than HDMI which is a better audio transport than SPDIF. And not all USB is created equal either. Using an audiophile grade USB card or a motherboard that has isolated DAC purposed USB ports will help, but at that point you need to have sunk a lot of coin into your DAC, speakers, etc to notice that difference, and it only matters to hardcore audiophiles.