Yes, but it is on it's limits... Especially the GPU is as absolutely minimum level.would it actually be capable of doing what I want?
This is not really the truth. Theoretical you may be able to play 4k Video, but only on low bitrates and especially not with h265 codec (the standard codec for 4k).specifically mentions being able to support 4k video.
I had a Core i5 from the same generation than the N3050 with a GPU some levels above. Even this i5 is not really able to play h265 4k content without blowing the CPU over the limits.. The GPU did not support h265 so everything needs to be done by the CPU. The Core i5 I own is many times as fast as the Celeron N3050. And even the Haswell i5 is pretty much as it's limits while decoding a h265 4K video... I really doubt it would work with an average 4k video on a N3050. only with specially adopted (transcoded) ones...
Drivers for Win 7 are no longer be available for many things, not only NUC... So you may be forced to use Win 10 sooner or later nevertheless...
By the way, Im absolutely sure Win7 can be installed on this device. No special hardware is installed on those systems. Aside this you always can install Win 7 via USB 2.0 ports. The NUC 6cayh has two internal USB 2.0 ports so you only need a cheap (less than 1 GBP) connector. Probably you will have one laying around from an older computer somehow. I'm sure I will find at least a dozen of them, if I dig deep enough into my storage..