Articles – ‘I Had Headache’ or ‘I Had a Headache’?

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.

Unfortunately, I don't know if there's any solid rule you can use to explain why "a good knowledge of English" is correct. You are right that "knowledge" is an uncountable noun, but in this particular usage the idiomatic way to express it is with the "a". It's possible that the idiom distinguishes between "a knowledge of English" and other kinds of knowledge, or other degrees of knowledge, rather than just talking about the amount of knowledge.

Still, again, it's hard to pinpoint which nouns work like this, and which don't. Also, "knowledge" isn't the only uncountable noun where this applies:

I hear you have a good intuition about the future

I hope you have a clear understanding of what I mean.

They're trying to find someone with a good insight into the stock market.

It may be yet another case where you just have to memorize these exceptions as you come across them.

Anyway, as mentioned in the above comments, the given answer to the second question might be OK in some dialects, but "We're having terrible weather" is also correct.