'Not a lot'

Birds (or sightings of them) have grown 'fewer' (Answer 2); but the number has grown less.
It always feels odd to come across such a phrase as 'growing smaller' or a sound 'growing softer', or a vehicle or process 'slowing down quickly' (or indeed 'speeding up slowly'), but sometimes these do make remarkably clear sense. Perhaps you have similar clashes of meaning when you express such concepts within your own language.

... i.e. she had many problems, and the one we're currently considering was the one that was least important, or most minor.
'Best' (Answer 1) really wouldn't make much sense here!

In this case, the adverb 'less' is modifying the participal adjective 'traveled'; it doesn't mean that 'less people traveled by it' ~ although it amounts to much the same thing, that would be inaccurate English (it should be 'the road fewer [people] traveled by'!)

There were at least that many, and they were countable!

This means that there may be no correlation between the extremes ('cheapest = best').

Answers 1-3 each seem to rhyme, appealingly (especially No.3 with its suggestion that 'well-bred' / well-mannered people don't 'go on and on' talking about a problem) ... but in fact No.4 is the correct ending. It aims to suggest that the more quickly people stop discussing a problem, the sooner matters will be forgotten and revert to normal.

Answer 2 is correct since we want the smallest quantity of countable things (in this case, vacancies : i.e. empty rooms, for which the hotel is paid no money).

The items are countable, so 'less' (while broadly clear for meaning) would be sloppy / inaccurate English.
Certain shops and supermarkets may display signs with the wrong wording, but that doesn't make it ~ or them ~ right!
As and when you ever see 'wrong' signage in English, do ask any sensible English-speaking friend and check whether the mistake is in the sign itself, or whether you've misunderstood something. Sometimes you may be 'at fault' yourself (but hopefully, will learn from it); sometimes you may be able to pride yourself quietly on knowing your English better than a native writer!

The workers are countable; we suppose that '£9/hr' is also countable, but Answer 3 remains the most appropriate English version to express this idea.

The 'number' is itself uncountable (!) so the correct phrase is 'the least number'; but the points, obviously, are countable so they will be 'fewest'.
All the other Answers are either illogical or grammatically wrong, on indeed both.