Do we always write 'W' capital in Western society, Western countries, Western culture, Western ethnicity?
Choose a particular style guide and stick with what it says. But be consistent throughout one paper or article or post or blog. But style guides and usage changes over time.
The Chicago Manual of Style prefers the "down style" (8.1, 16th edition) that is, to keep capital letters to a minimum.
It goes on to say:
8.46 Regions of the world and national regions
Terms that denote regions of the world or of a particular country are often capitalized, as are a few of the adjectives and nouns derived from such terms. The following examples illustrate not only the principles sketched in 8.1 but also variations based on context and usage. For terms not included here or for which no suitable analogy can be made, consult Webster’s or an encyclopedia: if an otherwise generic term is not listed there (either capitalized or, for dictionary entries, with the indication capitalized next to the applicable subentry), opt for lowercase. Note that exceptions based on specific regional, political, or historical contexts are inevitable and that an author’s strong preference should usually be respected. [16th edition, emphasis mine]
As you have experienced, consulting "Webster's" (or other dictionaries) and reference works sometimes clouds the issue. So I think you are justified to use whichever, Western or western. Just be consistent and be ready to back up your choice, if asked.
The Chicago Manual of Style gives these specific examples:
the West, western, a westerner (of a country); the West Coast; the West, Western (referring to the culture of the Occident, or Europe and the Western Hemisphere); west, western, westward, to the west (directions) [16th edition, emphasis mine]
And the APA has an article that is "an example of the use of bias-free language in scientific writing."
In it, one finds 'modernism' defined as "Westernized changes involving the adoption of new lifeways that include innovative and culturally different beliefs and behaviors" [my emphasis].
So, 'Western' appears to be in vogue for geographic and, I would add, Socio-Cultural-Politcal(?) entites as in 'the West,' 'Western powers', 'Western decadence'.
I guess the BBC prefers an even greater 'down style" when it says "western countries."
Perhaps this is in contrast to 'Western countries', which refers to countries included in 'the West,' the Socio-Cultural-Politcal(?) Entity that includes Japan.
Also, is it incorrect to write both the first letter capitals? 'Western Countries' in the middle of the sentence?
If you have a phrase that needs both words capitilized (i.e., a proper noun), then do so: 'Western Hemisphere', 'West Coast.'