Might not be of any help but worth checking this thread out:
https://communities.intel.com/message/232631#232631
I found you have to sometimes try the mini hdmi output, or you can try hitting f2 when u switch on the nuc and see if it goes to bios u can try to set the correct display output instead of "auto" like in that thread suggested.
I certainly found issues when connecting my normal tft seemed the display worked always first time when it was connected via mini hdmi to hdmi on a LG TV set oddly.
Also to get that nuc mobo out check this vid:
https://www.youtube.com/watch?v=eUaTrDywwUc
shows the guy removing those 2 screws holding the nuc mobo down and even he was lost and baffled why it just did not come out. He said he had to whack to top of the case a few times and then it came out! see the vid and he explains this, I think the bios reset is located on the underside of the case? not the best and easiest place to reset it really. I did notice some cmos perhaps password jumper on the top mobo part but it did not do much for me.
Don't forget to try hitting F2 to enter bios when you first switch on the nuc sometimes that jump started a pic on screen for me also. Also remember Uefi bios is quite fussy if you leave usb devices or pen drives it sometimes loads that up instead and goes blank... I found legacy mode and reinstalling windows 7 more better and issue free, you may need to play around with bios settings since legacy seems to like sata ssds more especially if you dual boot with win7 and win8.