If I'm understanding your post correctly, as shown in your mapping dialog, you map layers to entity types.
You don't map SW layer to DWG layer.
So, once you've mapped one entity type to a layer, you can't map the same type to another layer.
You're trying to tell SW to map some notes here and some notes there. There's no provision for SW to distinguish within an entity type.
SW just complains:
This is why all dims end up on one layer.
Not sure if I explained correctly, but can you have notes on 2 different layers or does Solidworks only look at notes and convert all accordingly
Not sure if it makes a difference or not, but the one layer is in the sheet format, and it still seems to change it all to the dim layer.
SW doesn't map layer to layer with the mapping dialog.
Only entities to a single layer (since once you've selected an entity type for mapping, you've can't pick it for another layer too).
As a suggestion, try exporting to DWG without the mapping file and see if it's closer to what you want: a layer to layer conversion (which is what will happen).
In save as DWG dialog, click options and uncheck the mapping checkbox. Continue with the export.