Just for info - after the latest firmware release my 2012 mini has shown no issues when using hdmi (which I use a lot as I have my Toshiba TV hooked up as well as my Ilyama monitor through miniDP).
I did have to calibrate the screen after the firmware as the colours were all washed and it seemed that contrast was knackered too, Once I did the calibrate it was like having a different monitor - the hdmi output was crisp, clear and colours were strikingly awesome.
I suggest anyone having issues still to check for the firmware update, and then rn the calibration - it seriously is worth the time.
As for fan noise - mine is a 2.3 i7 Server with 16GB Ram (3rd party) and 1TB HDD wih 240GB Intel 530 SSD. The only time I hear the fans is if I am using my Windows VM with Corel Draw (this seems to be intensive) and also when I am transcoding video files - although I'd say most times I transcode it runs the fans fairly low.
My experience with the i7 mini is a good one now the firmware fixed the hdmi dropouts. I was miffed at first when it seemed to still have rubbish colours with hdmi cable, but I followed the calibrate and WOW fixed.
And as a footnote to re-iterate whats been said here before: HDMI is digital. a £1.99 bay cable will do the same job as a £99 Harrods one. as long as the cable is not faulty; there is no difference.