I took a closer look on 1 and 3 and couldn't make either work. In case some other modder wants to take a look, I will list my thoughts on each:

1.  AnomalyDefs.xml, does have an unused (or rather all anomalies use the default) option for an image, but this is for the dialogue after surveying it. AnomalyArtDefs, does only seem to hold entries regarding the 3d textures, similar to how ship colors are defined. The image file for the strategic anomaly icon is in "Galactic Civilizations III/Gfx/StrategicIcons/StrategicIcons.dds", which is a file containing all strategic icons, including ships, planets, etc.
2. I don't think it is possible on planets/starbses/etc. specificly. Didn't think about it much though.
3. You can easily add a copy from the colonization tech for example, which has the tech tree itself as a prerequisit. What the game however does with it is the following: It takes the first tech with prereq techtree into the colonization button, the second into the engineering button and so on. This means the fifth branch won't have a button to click on. So you would ahve to add a button in some TechSomething.xml in "Galactic Civilization III/Screens", which is not conventional modding anymore and would ahve to be refreshed with every patch.

I just checked in the schema folder to find any hints in enums, but no. I guess it's coded and you can't change it without braking the game.