Learn English – Omitting articles in nouns – prepositions; after; to; before; from

The general principle here is just that the focusses on a specific instance of something, whereas a implies one among many possibilities.

Thus in #1, the reduction of mercury pollution implies either that it's just one example of a type of pollution that could be reduced, or that mercury pollution is a clear example of things you might spend $1.5 billion on. But it's optional regardless of the exact sense intended.

In #2, if the hours of darkness are few implies we're specifically talking about how many hours of darkness there are at the time/place of fasting. There's no special grammatical rule applicable here, but idiomatically I'm sure most native speakers would include the in such contexts.

In #3, a range of techniques simply implies that the writer isn't concerned with exactly which range of techniques he's talking about. If in fact he meant the particular techniques actually used by his company, for example, he could have said our range, More rarely, in contexts where he's thinking of the complete range of possible techniques, he might use the definite article. But it would be generally considered "ungrammatical" to have no article at all in this construction.

Note that these are very fine nuances that won't normally apply anyway. In all the "articles" I've looked at above, and in comments below, any of definite | indefinite | zero article could be used, and it's unlikely many if any variations would strike most people as "odd".

In the first example, Doctor is being used as the name of the person; the doctor is more of a descriptive phrase. It's short for Doctor <his name>.

tend bar is a set phrase, it's a synonym for being a bartender. It's also similar to the way other people describe their work: a mailman could say I deliver mail, a programmer would say I write code, a garbageman would say I collect garbage, and a composer would say I write music. These are all using the noun to refer to the general concept, rather than any specific item, so no article is needed. You would add an article when you need to be specific, e.g. I write the music in TV commercials.