I was not able to perform the download at your link. Instead, I visited the touch_terrain site and approximated the location. The modifications allow for a thicker base, which I selected to 4 mm arbitrarily, eliminating the non-closed (probably) as well as exaggerated the vertical scale to 4x.
The result is satisfying and appears to have a minor validation problem surrounding the perimeter of the base, but no problems with the terrain.
If adjusting the base thickness does not resolve the problem you experience, consider also to increase the overall size of the model and use the tiling function to segment as needed.
For the roadway application, you can create an SVG of the map at appropriate scale. Thin lines will be problematic, but the thicker lines can be boolean-subtracted from the STL file. It will be necessary to extrude the SVG to sufficient thickness to impact the terrain STL in order to create the trenches.
It might also be possible, but far more complex, to use a feature in Fusion 360 that overlays an extrusion onto a surface in a uniform manner. Angus from Maker's Muse has a video explaining the process, but it's a pair of primitives, not a complex terrain and may not apply.
With the permissions corrected, the file downloaded without complications. I note that the composition/analysis is in excess of 140K triangles (Meshmixer) and could create problems with some programs. Fusion 360 will either refuse to function or provide an alert.
I reduced the mesh using Meshmixer set to 70 percent, resulting in no apparent loss of detail, but fewer than 40K triangles. Tinkercad also dislikes high triangle count objects. The reduced file will be available for download for a limited time.
Image from released (reduced) STL file.
But wait, there's more. After some additional thought, with consideration for your constraints, I think I have a workable approach. I'm not sure how you'd approach this first step, though. I used Windows 3D Builder to repair the broken base. Meshmixer shows the flaw but does not repair it. I suspect other online repair facilities will resolve this simple problem. I could have tried Meshmixer (plane cut) and certainly Fusion 360's plane cut, but only thought of that as I type this addendum.
Once repaired, I saved your map image and traced it in a suitable program to create the SVG file. Both in Inkscape and in LightBurn (a laser editing/burning program), the tracing went well. LightBurn allowed me to confirm that the SVG generated had closed shapes, but Tinkercad would not import any variation I created. I used Inkscape's Path to OpenSCAD which previews fine the extrusion to 10 mm, but it would not render, complaining of non-closed shapes.
There are "speckles" all over the map, even though I enlarged it to 200% scale to reduce that aspect of the lines. I suspect this will be problematic and suggest to locate a better resource, to eliminate excessive detail in the map.
Once you have that solved and can extrude your map, scale and align it to match the terrain STL which has been repaired. Create a duplicate of the terrain, also properly aligned and elevate it. The object of this step is to use the terrain to carve away the extruded map. If the boolean subtract does not destroy the terrain, great, otherwise, lower the extruded map into the terrain to the desired depth and perform another subtract.
This should give you the channels you require for painting.