Here are two definitions of the word "down" from TFD.com:
Down a. Toward or in the south; southward: flew down to Florida.
Down b. In a particular part of a country: down south.
According to definition b., can I say "I'm down north right now" (down meaning a particular part of a country which is north)?
No, it is not idiomatic to say "down North".
It should be "up North". The metaphor comes from the usual orientation of maps. "I flew up to Maine", for example. The word "down" is only used with areas of the country that are southerly.
We don't really have anything similar for East or West. Perhaps "out West" or "out East" would work. "I flew out to Oregon" seems reasonable.
"Up North" and "down South" also work in Britain.
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.
As mentioned in this answer on English.SX, sometimes people say “down North” in places where the rivers run downhill to the north, such as arctic Canada. Examples include the book title, Down North to the Sea: 2,000 Miles by Canoe to the Arctic Ocean. “Down North” is also sometimes used as a proper name, including for a region of Ireland.
Nearly all the examples I found use “down north” in the literal sense that the destination is lower in elevation. It would not surprise me if people somewhere use it in a metaphorical sense. I could imagine someone saying “down north” as a shorthand for “downtown, in the city north of here,” or some stuck-up person going “down” to the bad north side of town.
Definition b in this case is highly idiomatic. The example given, "down south," is the pairing that most English speakers would use most of the time. There are words other than south that could be used with this sense of down, but only in very localized circumstances.
For example, there is a part of the coast of the state of Maine in the U.S. that is known as "down east" Maine. But if you were in the western end of Massachusetts (a nearby state) and told people there that you were going "down east" to Boston (which is in the eastern end of Massachusetts), they would think this was a very strange thing to say.
There may be other examples of "down _____" that are applicable in other locales that I do not know about. But in almost all places where it makes any sense to put a compass direction in the blank space in that phrase, "south" is the only word that is suitable. The definition as stated seems to imply that "down south" is only one of several possible examples that might commonly be used, but this is misleading.
As has already been said, the common phrases are down South and up North.
In England, owing to railway terminology, up and down can mean towards London and away from London respectively. So, even someone who is located north of London will say: "I'm going up to London".
For East and West, we often use over in statements like: "I'm going over to Ramsgate", (but not just over East).
The directions you hear in use in English are meant as map directions, relative to the middle of the map. When you hang a map on the wall, the standard is North is on the top (which puts South on the bottom, West on the left side, and East on the right side).
So, points far away relative to the center of the map (or perhaps to you conversationally) are "Up North", "Down South", or "Over in/to/at" for points East or West. This is to help those of us who visualize a mental map in our heads for finding our way about.
In many parts of the US, especially in hilly or mountainous regions, people often use up and down to correspond to difference in elevation regardless of compass direction.
For example, you could say, "I'm going down to the store," when the store is downhill and north of your current location.
Here's another common one: "We're going up on X drive," referring to a park located on top of a mountain.
All depends on where you are. There was a time when everything moved by sail, and the fastest and most reliable way to get anywhere was by going downwind. That's how we get going "Down East" for Maine, because east is downwind of the prevailing westerlies. But modern Maine has succumbed to conformity and today, to all but die-hard yankees and sailors, "Down East" means a popular magazine and not much else. However, in Cape Breton, every day locals still refer to going north as going down. Similarly, "below" means to the north (eg: Cheticamp is below or "down from" Inverness), and when something is "Lower" in a place name, or even generically as in "the lower settlement," it means north of someplace. It drives American tourists bonkers who are always getting lost, having interpreted the directions they were given in an upside down fashion.
In Canada (from the Great Lakes to the east at least), saying "down east" (as opposed to "out west") is common.
It's likely because the Great Lakes and the St. Lawrence River flow from west to east across half the country, so if you are in a boat going downstream, you're heading east.
I imagine "down north" or whatever it's equivalent was, would have been common in lands around the Nile River, which flows from south to north.
