So I set up ambient light and a skybox in the Render settings, then press Play and the Render settings (which are still displayed) change to different (default?) values. The skybox disappears, the ambient light changes.
Is there some field that controls this or might some third party script be doing it?
Skybox is set on a per-scene basis. Try creating a new scene, then making sure Skybox in Render Settings is set to None. If you hit play in the new scene with no scripts and it changes, it’s probably a strange Unity bug. If it stays set to None, some script is probably changing your Skybox and other settings.
To find the culprit, you could search for ‘SkyBox’ through all your scripts (pretty easy to do in Visual Studio with CTRL+F. Not so sure about MonoDevelop).