Is it possible to do a clean reinstall of Skyrim (non-Special Edition) via Steam with DLC?

If you have purchased the DLC for Skyrim, it should be associated with your account and should still be accessible even if you reinstall. You should be able to see a small table of the DLC you have for Skyrim and whether they're installed on the game library page. You can check this to make sure when you reinstall Skyrim that the DLC are successfully reinstalled