Say, I flash it with the STRIX BIOS for the sake of OS X; will that change user experience of the card on Windows or during demanding games like Watch_Dogs? Can it lead to hardware failure, overheating or any such issues?
Yes, it might.
I flashed my ASUS GTX 750 Ti OC 2GB DDR5 with
Palit's 82.07.32.00.60 BIOS. After that I was able to boot into OS X using HDMI. However, whenever I launched a game in Windows, green flickering pixels appeared and the game crashed instantly. I thought one reason was that the core clock rate of Palit's BIOS was set to 1202 MHz but my card's base core clock rate was 1072 MHz. So Palit's BIOS was overclocking my card, making it unstable.
Therefore, I tried to modify Palit's BIOS using a tool I downloaded somewhere. From the tool, I learned that Palit, ASUS and Gigabyte's 750 Ti BIOS all have different settings for fan speed control. I changed all values (fan speeds, core clock rate, mem clock rate, etc.) to values presented in the original ASUS BIOS and flashed the card again. I was able to boot into Windows, but NVIDIA driver could not recognize the card. The driver was not loaded even it was installed.
Then, I flashed the card with Gigabyte's BIOS, which had a lower clock rate, but the card was bricked. Luckily, I was able to re-flash the card with the original BIOS.
As I remember, ASUS STRIX 750 Ti series have higher base clock rates than normal ASUS 750 Ti. I think it's likely that the BIOS will make your card unstable.
In conclusion: Flashing with third-party's BIOS is not a permanent solution to the "missing HDMI" problem.
Update: ASUS STRIX's VBIOS 82.07.32.40.47 is much stabler than Palit's VBIOS with my card. Now, I can play all my PC games without any issue.