The project would be a turnkey video player: plug it in to the hdmi display and power, and it plays a prepared video list in a loop. No user controls or anything; brute stupid to use in the field.
Is the pi-zero fast enough and/or its GPU capble enough to play full-HD video? Note that it’s not some arbitrary files but fixed prepared files, so I could be sure to use a suitable codec and compression options.
Yes. As far as I can remember, pretty much every Pi I've ever tried to play 1920x1080 video on has worked, up to and including the very first generation boards. You can see a brief sample of the Zero playing back 1080 resolution footage from Big Buck Bunny using Kodi on the KordKutters Youtube channel. Looks fine to me, although the presenter notes a little slowdown during add-on updates - you shouldn't have this issue if you're only playing back footage.
If your end goal is just 'build a Pi that plays looped video' it may be worth a look at the ready-made image provided by MP4Museum. It doesn't do much else, but it apparently does that well.
Yes it is. And it's also worth mentioning omxplayer, a neat command line video player with lots of options, see this elinux post and this RPi documentation.
Install it with
sudo apt-get install -y omxplayer
If you want to compile omxplayer from source, follow this readme on github.
omxplayer is deprecated since Jul 2020: github.com/popcornmix/omxplayer