My personal experience:
Running VMs, no problem. As long as the VMs are not CPU heavy
Run dual monitors: as long as you pick the correct NUC model, check the maximum resolution supported for multiple display. (Intel® NUC — Setting Up Multiple Displays) For the new NUCs that just came out this year you can check the tech spec of each to figure out what resolution they support (Intel® NUC — Technical Specifications) Different NUC has different display connectors, some has only 1 HDMI output, some has 1 mini DP + 1 HDMI, some has dual mini DP. If your monitors do not have a mini DP or DP connector then you could face an issue if your monitor has higher than FHD resolution (requires active DP to DVI dongle, ~ $100) otherwise you can use passive DP -> DVI dongle (~$5)
Play games: If monitor is not high resolution (above FHD) light games should be OK
Generic computer stuff: no problem
Programming: no problem
Basically, NUC can do everything a decent laptop can do. It is essentially a laptop without a screen and packed into a tiny square box.
Please refer to the tech spec for the specific model you are interested in buying to confirm below:
VT-x: I think all of them support it.
VT-d: the cheapest one based on Celeron does not support it, maybe i3 as well, all other NUCs should support it
64 bit: all NUCs support that, I think.
If you are looking for NUCs with dual mini-DP, there are only two models that I know of:
Ivy Bridge: DC53427HYE