I want to create a map where provinces / countries share borders but are distinct objects that I can colour in and modify. 

An example below of what I'm talking about. 

One way is to draw all borders with a non-dashed stroke - combine them when done and the use of the fill-bucket to fill in coloured shapes in the area of the "countries" with enabled grow factor to make sure color fill grows at least to half of the thickness of the stroke (or add a stroke of same color to the countries) - then raise the border-lines on top - and adjust dash-lines. Example attached.

Thank you so much. Do you mind explaining how you move the border lines to the top? Also, would it be possible for you to send me an SVG of the example you posted? Thank you for the help.

The image above is an Inkscape file.

On top means in the objects hierarchy:

Thanks. It seems no matter how much I change the grow factor and stroke the fill doesn't cover my province, and it also leaves these ugly paint "blobs" outside the border.

The fill tool works zoom dependably: try to maximise the area you want to fill to screen/Inkscape window size. When "Grow"-factor don´t work add a stroke width.