This is not a question whether DR could do it or not, but rather whether the id Tech engine can do it. Short answer, it cannot!
Hum, i don't agree, i saw nothing on that video that made me say, yes this is pretty much impossible in idtech 4, what the artist was doing imo was just vertex painting or texture blending, two things you can render in idtech 4 just fine, just not inside DR (you can place decals but that is technically different from what i'm saying).
I know painting inside DR would be a fantastic thing to have, even tho I don't know how difficult it would be to implement, i assume is not a easy task, DR is also being worked by a single person, Greebo, on his free time, so unless some other blessed soul starts helping, i don't think this will happen anytime soon, IMO for the time being you guys can just export the surface you want to "paint" from DR has a obj, open that on blender/Modo/Maya and paint there, if you know how...
Btw Judith, unless the TDM team removed it, idtech 4 has a GUI editor (from Quake 4) just type "editguis" on the console, just know this, it has the tendency to crash, so save often, and the gui preview inside the editor is not exactly how it will render ingame, specially if you put animations on it, is good for static GUI's tho.
This is personal opinion but i think you really don't need a GUI editor if you are making GUI's, there's some console commands that imo make working on them a breeze.
testgui guipath (if you put this on a shortcut or .bat file, for example "name.exe +set r_fullscreen 0 +set r_multiSamples 0 +set r_gamma 1 +set r_mode 6 +set vid_restart +testgui /guis/name.gui") it will open the game in a window and immediately open the gui, then just use "reloadGuis" on the console to see all the changes you make to the gui code, in almost real time without the need to start the game again or reload a map.
Another nice command is "gui_debug 1" this makes the bounding boxes of the various windowDef's visible (just like the gui editor) and also shows the mouse screen coordinates, good to know the screen coordinates to input into the windowDef's.
Hope this was helpful.