I've been using a NUC for a year and a half, they run perfectly off 12v (if you look in the Intel spec PDF, it actually lists 12-19v).
Unless you buy it already fitted out by a third party, you do need to add RAM and storage, but neither of those need be expensive. The bottom of the machine lifts off after undoing 4 screws. The RAM and storage secure in very easily.
I use my NUC headless and access via RDP - this is a built-in function of all Windows OS for aeons, no additional software required. Since RDP is such a prevalent protocol, apps for non-Windows platforms are common, some even written by Microsoft (not that that in itself should add any legitimacy
)