The only spec I can find refers to 'Rich Text edit control' and 'predefined dialogs'. That appears to rule out a native code solution (in BBC BASIC or anything else) since by definition it won't be using any 'predefined' library widgets.
What challenge? I generally don't take an interest in challenges (I leave that to my users). The Liberty BASIC GUI editor FreeForm will be hard to beat (it runs in my LB Booster too), if existing programs are eligible.
Yes, for example, and the rest. The GPU is good at that kind of thing as well as blitting textures. You could compare the speed of SDL_RenderDrawLines or SDL_RenderFillRect when using the GPU compared with software rendering.