I ran Skyrim (and Oblivion) in wine for quite a while, with a large number of mods (200+). I haven't tried to run it in wine in a couple years. When I did, there were three issues I tended to run into:
Mod Management
Nexus mod manage always failed for me in wine (dll and .net dependences) and mod organizer uses a virtual filesystem which left wrye bash--even it could be flakey. I ended up using a windows VM to run wrye bash with its own skyrim install. I synced the VM's install with the wineprefix's install with ssh.
I tried to use various content repositories for mod management, but none seemed to handle various file issues well. Especially since bethisda uses file dating to organize file overwriting during loads.
Filenames/Calls
Windows isn't case sensitive, but linux is. Wine has a translator which tries to match calls to windows files with linux files, but a heavily modded skyrim can have thousands of files to search and pattern match. At first it doesn't seem like much of an issue, but consider the case where a mod author writes a script which calls a common file and uses a case which doesn't match ANY other mod author. Including the developers. (Even the developers aren't consistent with file names.)
Now toss in how the graphics files (models, textures, etc) all have internal references to the other files needed... it can be quite a mess.
The result is randomly missing textures, models, etc. Sometimes the game will just be ugly, other times it crashes because a script isn't able to load what it wants. When problems occur is random, sometimes wine will be able to pass the right file, sometimes not. Remember, 10s of thousands of calls are made during a load.
Mods which packaged in BSAs tended to work best--but these could make for some surprise conflicts.
(The explanation in regards to the file matching may not be altogether accurate, I'm working on memory from a couple years ago. Also, considerable improvements could have been made since then!)
Dlls/apps
Any mod which has its own dll libraries or an executable can be an issue. Much of the reason I used a VM for animations and body slide. Some of these could work in wine, some may not, some may change from version to version--all depends on what they require.
SKSE brings to the table additional weirdness. I had several mods which used SKSE which worked quite well. Others (usually ones with their own dlls) didn't.
The Fun Stuff
The most important thing to remember when modding in linux is to keep it simple. If you love modding, like I do, then doing it in linux is Hard Mode. When troubleshooting an issue it's difficult to tell if it's an script, mod, or wine problem.
It can be quite a headache.
My linux box is 8core, 3.7ghz, 32gb ram, sata3 drives, etc. Pretty solid. My windows comp is a dual-core, 2.2ghz, 8GB ram machine with sata 1.1. Sjyrim runs better on the windows box, and it's easier to manage everything with Mod Organizer, bodyslide, ENB, etc. At last count, I had almost 350 mods shoved into my skyrim install.
So, it's a bit of a mixed bag. Most mods will run fine in linux. Some will drive you insane* trying to figure out what's wrong.
Though I'm tempted to give modded Skyrim in wine a try again, especially with staging...
*
in this case, insane means "back to windows"