"The" for superlative referring to more than one object

Solution 1:

The 1st and 3rd examples are correct, because when talking about superlatives, you are effectively talking about something specific, so the definite article must be used, whether the thing in question is singular or plural.

A noun can be omitted after "the best" in the 3rd example, but it is implied, i.e.:

Those apple pies were the best [apple pies].

Those apple pies were the best [I've ever tasted].

Solution 2:

Using the article doesn't depend from the presence of best in the sentences.

I prefer "the best countries to live in", in the same way I prefer "the countries I would like to live in".

For the second example you made, in both the cases the article is used.

Solution 3:

I think there may be some confusion embedded in your question as it relates to comment on the wiktionary article. I think the usage they are alluding to would be of the following structures:

• "X is the tallest"
• "X is tallest"

It seems that wiktionary is pointing out that using the definite article with a superlative is preferable to a) not using an article at all, or b) using an indefinite article such as 'a' or 'an' considering those do not make sense. So yes, in your examples, #1 is the best, but I am guessing you already knew that.

Solution 4:

Both of your original sentences are correct, though with different nuances.

The best countries to live in are X, Y, Z.

This enumerates those countries that are best to live in. Or:

The best countries to live in have lots of cheese.

This means that, out of the finite definite set of best countries to live in, they all have lots of cheese. This, however:

Best countries to live in have lots of cheese.

Sounds more like you're giving guidelines for finding best countries. That is, look for countries with lots of cheese, and you will find some of the best among them. Maybe it makes more sense with another example:

Best practices include documenting your code and having large test suites.