Society without an article is the abstract concept of society:
Society as a whole needs to take responsibility for these problems
Society with an article refers to a particular organization
I keep my money in a building society
I am a member of the amateur dramatic society
Something is generally countable except for
- abstract concepts like peace and fairness
- activities like swimming
- substances that are infinitely divisible (water, sugar, leather, soap cheese, etc)
Note that you can still use a plural for something that is uncountable, to
indicate there are multiple types of it:
I tried several cheeses at the food fair (meaning several types of cheese)
Going back to your examples,
Society doesn't pay enough to old people.
This is society as an abstract concept.
The society doesn't pay enough to old people.
This would be a particular organization, for example a charity that helps old people.
Societies don't pay enough to old people
This could be either referring to several different organizations (for example charities) or different kinds of abstract society, for example in different countries.
I think that the first example has the intended meaning and the other two, while grammatically correct, probably do not have the intended meaning.
Best Answer
The short answer is no. The adjective does not change the syntax.
The long answer is that nouns can convert class from mass to count or vice versa. There's an old joke that if you have a really powerful machine that turns anything into powder ("a universal grinder") then you can make any noun from a count noun into a mass noun -- "that's a lot of man on the floor."
To go the other way, you don't even really need a machine -- whenever you are comparing multiple instances of a mass noun, you convert it to a count noun: "the sands of Texas are whiter than the sands of Florida."
As a result, you will find many instances of nouns that are usually mass nouns being converted to count nouns.
For your examples, I find "a soothing music" hard to parse, "a deadly pollution" makes sense (we are imagining several instances of pollution, this one a deadly one), "a hot tea" is fine (with food items that are mass nouns, we convert them a lot when we are ordering them at a restaurant, e.g. I'll have a tea), "a hard work" sounds terrible to me when describing the mass noun "work" (which means labor) but there is also a count noun "work" (a creation arising from a lot of labor, like the works of Beethoven) and here it makes sense to describe one as "hard."
But the point is that the adjective has nothing to do with it. It's just that the mass noun has been converted to count, and there happens to be an adjective.