I couldn't find a mod that did exactly these two things so I made one myself.

This mod removes quest markers and enemies from the compass. As a side effect, this also removes quest markers from the world map and local map (seems there is no workaround for this yet). I recommend using the mod Even Better Quest Objectives alongside this mod.

This mod does not remove any location markers from the compass or map. This mod does not remove the personal map marker (blue arrow) from the compass or map.

To install, copy the contents of this folder into your \Skyrim\Data\ directory. To uninstall, delete the file located in Skyrim\Data\Interface\Exported\, called hudmenu.gfx.

I'm currently using this with SkyUI and it is compatible. Haven't tried it with any other interface mods.