I have a few downloads in my game I no longer want, so I would like to delete those. Unfortunately the Delete-Button is gray and I can’t delete it.
Unfortunately it’s not possible to delete downloads directly from within the game. You will need to uninstall them through the launcher or delete them from your mods folder. See this for details: Remove Sims3Packs.
The Delete-Button ingame only works for ingame created content (which would be Recolors, Patterns and Tattoos).