While any machine that provides storage through the network can be a NAS, you'll probably want at least some level of extendability. Your biggest problem with OEM machines (think Dell, HP, Lenovo) will be s lack of SATA ports, in combination with not enough power outlets and not enough space to put the actual disks in the case.
That's the reason I usually build my own machines for those purposes, and depending on what's needed you don't even need high-end parts, at least for a "NAS-only NAS". My NAS works on a 2 core/ 4 thread Intel i3, which will be enough for the foreseeable future. But it also only provides storage, all the rest (self hosting my services) happens on another machine.
TrueNAS works well for me as a NAS OS. AS far as I've seen it also provides direct container support or virtual machines. It you want an all-in-one machine, that might be worth looking into.
All of the kind of files you listed currently live in a Nextcloud instance, although I also have a test paperless instance, especially for the scanned documents.
The only thing not in there are media files which can always be re-downloaded, I.e. files not created by me. Think music, movies, etc.
H.264 for DVD content is perfectly fine. H.265 will save a little storage, but that's basically it.
If you need to go outside your network it will suddenly be a lot more effort. I'd suggest a Wireguard tunnel, but in theory you could also open up the server to the internet. But you better know what you're doing in that case.
Transcoding is taking an already encoded file, e.g. in H.265 and "re-encoding" it to something else, e.g. to H.264.
This is usually done for clients that cannot natively play back the originally encoded files, or for reasons like bandwidth restrictions, subtitles, etc.
In theory you can get around that by originally encoding your DVDs to a format which all of your devices can play natively. Nowadays, on most modern devices you should be good with H.265. Best way would be simply to try: encode, copy over, play.
H.264 is supported by basically every not ancient device.
Remote streaming inside the same network is as easy as pointing the Android app to the server and logging in.
German.