Yes, DSM is the OS on all the synology nas enclosures. I've heard you can install it on custom built nas devices, but I don't know the details there, or if its easy to do or not. I would suspect its probably more difficult than not, just because synology is in the business of selling their nas devices more than anything. I have no idea how it would work installing it on 3rd party hardware at all, though.

As for synology moments, its an app that can be installed on DSM. Most of the additional apps are free (moments included), but off hand i know of one notable exception: Surveillance. You need a per camera license for their surveillance software, and IIRC every nas device comes with a "free" 2 camera license, but you have to purchase more if you want more cameras.

They actually have a pretty good ecosystem of apps on synology as well, including things like docker, plex, git, etc. that can all be installed directly on the nas itself and run as a service off of it.

It's worth noting that if you're buying the enclosures directly from synology, they generally don't come with any HDDs at all, you have to buy those separately. Not sure where you're seeing your "$200 for enclosure + 2TB", but i just wanted to put that out there as "make sure it actually includes drives if its through an official store or something" warning.