Sorry for the late reply. I have been travelling and struggling with really poor Internet. I switched to 9 so long ago that I don't remember the details of 8 for projections very well, so I can't help you much with this (other people on this forum no doubt could advise you better).
Keeping that limitation in mind, my gut feel is that your problem has to so with some small detail not being done right in the workflow, either in how the initial re-projection was done in 8 (try different options, like Preserve local values), or having to do with documented or undocumented details as to how the CAD system you're using imports projected info, etc. One reason I have that impression because you're describing not one specific instance but several problems (since you mention both raster and vector), and it's highly unlikely that a common bug is popping up in such radically different workflow. But it's very routine that issues in workflow cause coordinate system interchange problems between GIS and CAD packages.
Usually, the quickest way to find the correct workflow is by focusing on specific details for a specific example.
Start with a specific vector layer that is imported from a specific, original file. Keep a detailed log of your workflow for that layer, every dialog and every setting in every dialog from beginning to end, from the original file you downloaded all the way through to the specific file created for export. Next, keep a detailed log of your workflow for that exported file into your CAD program. If something isn't the way you want in your CAD program, repeat the entire process starting from the original file but making just one change in the workflow, for example, like changing one option / checkbox when you assign the projection, reproject, or whatever. Focus on getting that one file correctly into your CAD program and then you'll have a basic workflow example that will be easier to adapt to other vector or raster layers. I'd start with a vector layer since interchange to a CAD system can be a lot more complicated with rasters.
I couldn't follow in your description what, precisely, was done for a specific file as you refer to both vectors and rasters. In the example project, I don't know which of the layers in the Map are the troublesome ones.
It's also important to stick to a single package, to try to debug one workflow at a time. For example, if you export something from Release 8 and then re-import the exported file back into Release 8 and it doesn't show up in the same spot, that's something that's actionable entirely within the 8 universe and within the collective expertise of this forum. But if you export a file from Release 8 into myCAD Plus and in myCAD Plus it doesn't appear where you expect, well, then you're also debugging workflow in myCAD Plus. That could be two separate workflows to debug, with this community unable to help with myCAD Plus.
Especially with CAD systems (which are notoriously troublesome when it comes to projected data) there are all sorts of details involved in getting them to display projected information correctly. Pretty much every online forum for every GIS package has posts complaining about interchanging projected vectors or rasters with some CAD system (with posters dissing the CAD system...), and pretty much every CAD forum has threads about all the problems importing data from some wretched GIS system (with posters dissing the GIS system...).
A practical problem is that most CAD systems have no capability to import projection-aware formats like GPKG for vectors or GeoTIFF for rasters, so they have to use sidecar files as hacks. GIS systems usually can work with variations in coordinate systems that are basically equivalent, like using local offsets or scales, while CAD systems usually cannot handle such equivalencies. Hacks like using XML sidecars to convey projection information are not always a sufficient to tell the CAD system what it must do in such cases. You might have to use 8 or 9 to cast the layer into a form that is not just georeferenced in Manifold, but which also is in an equivalent form that the CAD system can understand.
When Release 8 exports a projected drawing to a DXF, it exports a sidecar .xml file. When 9 exports a projected drawing to a DXF it exports a .prj. You might try using 9 for your DXF exports to see if the CAD system you're using understands the .prj better. For example, the XML uses scientific notation, as in
... which might confuse a CAD system. PRJs arose as a tradition during the times of primitive GIS packages so they use simpler words, like one does when talking to a child:
... the CAD system might be able to handle that better. Most CAD systems that claim to be able to read projected info are better at understanding .prj, since that is a very old convention.