How to manage Skryim (Workshop vs Nexus)

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).

1. Yes, you can use both.
2. If you plan on using many mods or large mods, you should definitely use BOSS to ease the load order management.
3. There are always incompatibilities between mods, the source of the mod shouldn't matter. The only important thing to note is that Steam Workshop will automatically update your mods, whereas the NMM will not. Updates can cause incompatibilities, and often do. Both BOSS and NMM will try to alert you if there's an issue.
4. There are a lot more mods on the Nexus (over 30 000 vs. Workshop's 19 000) and most mods are only available in one or the other. The larger mods, though not the largest, are usually available in both, but anything depending on another mod is usually not found on Workshop.

Generally speaking, Workshop is a good place to start modding for the first time. Moving on to something that doesn't automatically update your mods, such as Nexus Mod Manager, is preferred as updates often break compatibility between mods.