I have a strange glitch. When I enter the game with certain saves, there appears several mammoths, a giant and some netches and they follow me around. Even in some interior cells. I deactivated all my mods even the DLCs and they still appeared. I tried starting a new game and I got them even in the wagon ride to Helgen. I finally got a new game to start without them but I have no idea why they appeared or why finally they didn't. I want to try reinstalling my mods but I'm a bit nervous. I would have thought starting a new game without any mods would always work but it took a number of tries. Why?

Try it that way:

First do a standard uninstall. (Control Panel\All Control Panel Items\Programs and Features-> Uninstall The Elder Scrolls V: Skyrim)

Next, delete your skyrim folder under C:\Program Files (x86)\Steam\SteamApps\common\

Then delete your Skyrim directory under Documents/ My Games after backing up your save files if you wish.

Then under C:\Users\[Your Username]\AppData\Local\ - Delete the Skyrim folder that is in it.
(You should enable to see the hidden files under windows to delete that folder.) HOW: (View - Options - Folder Options - View - Advanced Settings: Show hidden files, folders and drives)

Use Ccleaner after all this (http://www.filehippo.com/download_ccleaner), and install your game again, from scratch. !!!!

You don't need to uninstall Steam.

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Then start a new game, normally you shouldn't have any mammoths in your game at the beginning. lol

Install your mods, one every time, and see if you encounter the same problem.

I would begin with mods, that are not related to animals - use NMM.