I'm confused about what you mean by "the DLC with mods I made carried over." There is a difference between DLC and mods, both in how they function and where those files are located.
Game files, including official DLC (and expansions), are located in your Steam folders (usually a path like D:\Steam\steamapps\common\Sid Meier's Civilization V) while mods are located elsewhere on your computer (often a path like D:\My Games\Sid Meier's Civilization 5\MODS). User mods that are structured as DLC, like the Enhanced User Interface (EUI) mod (which I don't see on your mod list, but it serves as a good example), have to be installed in a manner similar to official DLC in order to work -- they will be installed in the official game files' DLC folder (usually a path like D:\Steam\steamapps\common\Sid Meier's Civilization V\Assets\DLC). User mods that are structured as DLC will load every game, while user mods that are structured as traditional mods are only enabled if you specifically enable them from the Mods submenu on the game's main menu page. So, the fact that Steam will automatically re-download traditional mods into your Mods folder should have nothing to do with your problems, since those mods just sit there, having no effect on your game unless you specifically re-enable them for a specific game.
So, where does that leave you? If you truly want a fresh installation, and suspect you might have a DLC mod or two lurking around, you can do the following:
1. Go to your Stream Library, right click Cv V, and choose Delete Local Content. That should clean out the official game files, but will leave in place the folder structure for any folders (like maybe the DLC folders) that have non-official content.
2. Go to where the official game files used to be and check whether there is still a Civ V folder structure at that location (e.g., D:\Steam\steamapps\common\Sid Meier's Civilization V -- obviously, your specific path may vary slightly). If so, you can either delete that folder structure manually, or rename the top level folder to include extra words (like "(Old)" or some such), so a new version of that folder structure is created when you reinstall.
3. Go to where the game stores your settings, save files, mods etc. (e.g., D:\My Games\Sid Meier's Civilization 5) and either manually delete those folders, or rename the top level folder to include extra words (like (Old)) to force the game to create new folders after your reinstall.
4. Then tell Steam to Install the game. Steam Workshop should automatically reinstall all of your subscribed mods, but, again, you don't have to enable any of those mods to play the game, so any buggy or conflicting mods should not be an issue as you try to recreate a clean installation.
