Solution 1:

The number of mods Skyrim can handle seems to be limited by an unsigned byte, meaning slightly less than ~256 mods. (this account says 252 accounting for Skyrim.esm, the Update.esm, and the Saved game)

If you need more, some tools such as Wrye Bash can combine mods together, but this means you will need to manage these mods outside of Steam Workshop (Nexus Mod Manager is a good option).

Previously there was a 50 mod subscription bug with Steam Workshop, but it looks like Update 1.4.27 fixed that issue.

Solution 2:

Oblivion and Morrowind where limited to 255 mods, so we could assume that Skyrim have the same limit too. That said, 255 is still a rather high number, so unless you're grabbing every mod you find, you should be fine.

As for compatibility, it really depends on the mod and the load order. Some mod may work without problem, others only in a certain order, and some others just won't work together. It is a case-by-case problem, so you should carefully read the README of the mods, and you should try using tools such as Wrye Bash and BOSS to keep them organized.

Solution 3:

it depends on the mods. Remember, mods are made by random people. If you're lucky, they do enough quality control on their own works, but most probably they don't care about other mods (unless they're really really famous). So recommendation, just use what you really need.

Solution 4:

Only subscribe to dynamic and constantly updating mods. For example, the dovahkiin hideout or midas magic, which keeps updating.

But for something like the paarthurnax fix or perfected weapon mods (that require no more attention), you unsubscribe but keep the data files so they remain in your skyrim but are no longer 'subscribed' as such.