‘Skyrim HD – 2K Textures’ mod installed via NMM not showing up in the game launcher

1. Yes, they both work at the same time. However, I prefer to use NMM as it provides a better way (for me) to download, install, and manage mods than the skyrim workshop. For instance, it gives you finer control at ordering the mods, this is especially important for the unofficial patches.
2. I don't use BOSS so I can't say either way.
3. You will ALWAYS have incompatibilities if you use mods together, no matter what you do. NMM does its best at identifying possible conflicts.
4. There are more mods available on NMM since the workshop limits sizes of mods(which is why falskaar isn't in the workshop), as well as not having any mods that uses the skyrim script extender. (SkyUI is my favorite 'mod' for Skyrim but uses SSE).

Mod Organizer and Wrye Bash will both display missing and present Masters for mods.

If you don't want to make the switch to MO (though I'd recommend it) and you don't need a bashed patch, it would still be worthwhile to download Wrye to just run every now and then to check dependencies.