Others have answered the (im)possibility of using "down north", but I think there is also a misconception on how to interpret dictionary entries based on your question and comments.
A dictionary entry records a word with its possible definition or definitions. However, it is most often not a complete description of the word's usage.
If a dictionary says word A has a meaning B, it means that the word A can mean B in certain contexts. It does not mean you can use A in its B sense in every context.
Down b. in a particular part of a country: down south.
This means that down can mean "in a particular part of country", for example, in the phrase "down south" = "in the southern part of a country/geographical area". However, you cannot use the word down whenever "in a particular part of a country" is intended. The entry is only a description of the word's meaning in some situations; even if down can only mean "in a particular part of country" in the phrase down south, the entry is still valid.
For example, dry in "dry wine" means "not sweet", but if you cannot describe a savoury pancake as dry just because it is not sweet.