Everything seemed fine except for general Tullius (and later Hadvar) being half-stuck in the ground during the execution. At that time, I thought it wasn't a big deal. Everything else went like it should until I entered the Helgen keep with Hadvar.
Inside, I saw http://dl.dropbox.com/u/35436007/ScreenShot4.jpg. There was almost nothing after this room, http://dl.dropbox.com/u/35436007/ScreenShot3.jpg with some other parts of the level far away.
I reloaded the autosave and the keep looked fine now. Until Hadvar tried to get to the room with Stormcloaks. He stopped halfway in the corridor and refused to move forward, no matter what I tried (console commands as well).
At that point I knew there was something really wrong with the game and decided to try unchecking the mods etc. Hadvar was still stopping at that point even without any mod loaded.
It seemed like he was already bugged in the save. I started a new game without any mods. Got to the part right before entering the Helgen keep and saved (everything was fine). Hadvar got through that point like he should and everything was ok.
I tried it with all mods loaded. After entering the keep the blue void was there. I was happy - I had a save suitable for checking which mod was messing up.
Now it gets strange. I was unchecking each mod until I found the one that was messing up, and I found it. But I decided to try and run the mod alone to make sure it's it and not some conflict. I was slightly surprised to see it worked fine.
Now I started checking all the mods again to find out when will it stop working. When I found the second mod, I tried the game with only both of them active. It worked fine again.
The process repeated a few times. After a LOT of time and game restarts, I think I have found them all.
Now, the strangest thing about that is that all of the mods have to be activated for the bug to happen. If I disable even one of them, the bug's gone.
The list:
Update.esm
http://skyrim.nexusmods.com/downloads/file.php?id=11803
http://www.darkcreations.org/forums/files/file/205-unofficial-skyrim-patch/
http://skyrim.nexusmods.com/downloads/file.php?id=8601
http://skyrim.nexusmods.com/downloads/file.php?id=10886
http://skyrim.nexusmods.com/downloads/file.php?id=11555
http://skyrim.nexusmods.com/downloads/file.php?id=9778
http://skyrim.nexusmods.com/downloads/file.php?id=11803
http://skyrim.nexusmods.com/downloads/file.php?id=12369
http://skyrim.nexusmods.com/downloads/file.php?id=12024
http://skyrim.nexusmods.com/downloads/file.php?id=8971
http://steamcommunity.com/sharedfiles/filedetails/?id=17807
http://skyrim.nexusmods.com/downloads/file.php?id=13268
http://skyrim.nexusmods.com/downloads/file.php?id=11803
http://www.darkcreations.org/forums/files/file/205-unofficial-skyrim-patch/
http://skyrim.nexusmods.com/downloads/file.php?id=8601
http://skyrim.nexusmods.com/downloads/file.php?id=10886
http://skyrim.nexusmods.com/downloads/file.php?id=11555
http://skyrim.nexusmods.com/downloads/file.php?id=9778
http://skyrim.nexusmods.com/downloads/file.php?id=11803
http://skyrim.nexusmods.com/downloads/file.php?id=12369
http://skyrim.nexusmods.com/downloads/file.php?id=12024
http://skyrim.nexusmods.com/downloads/file.php?id=8971
http://steamcommunity.com/sharedfiles/filedetails/?id=17807
http://skyrim.nexusmods.com/downloads/file.php?id=13268
Now, the solution for that seems to be really easy - just disable one of them. I think this is really strange though, the mods don't really seem to be related at all and it looks like they trigger some kind of engine bug. For some reason they completely break the game when together.
I repeat, the bug is always there when all of these mods are enabled. It doesn't matter if they're the only ones or if there's more mods.
Anyway I thought that was interesting and hope maybe some of you have any idea why would this happen - the mods don't all seem to be related. Most of them are relatively large though. I might have triggered a really specific engine bug.
This is on patch 1.5.26. Doesn't matter if I run it with SKSE or without.
TL;DR version - Bug occurs with specific mods loaded, doesn't if you disable just one of them.
PS. I wish we had TES5Edit. I could check what the mods have in common and I don't seem to be able to do it with the tools we have right now.

There are academic fields of research that study this topic in various forms and different areas. My own current field, technology studies, is one such area. People who understand complex systems and their issues may point out a problem, but they'll normally mention that it may be a particular mod or it may be some form of interaction with a mod, other mods, the official software, and their specific hardware and software setup. Basically, there are many other elements that could be the cause of the problem or rather there are many interactions that could be the actual cause. Still, it's important to inform mod authors because the problem might only be their mod(s) and they won't know to check unless they are informed.