Why usually is the word “halfway” used with “down” rather than “up”?
I doubted that. I can't imagine that there is some kind of rule to use down rather than up. It's possible, but I didn't see one. I looked to Ngram.
So the usage looks roughly even, at least in recent years.
One reason you might use one over the other is perspective. For example, in a larger context (like a story), if you want your audience to visualize themselves looking down at the nest (fixed, not moving), then you might use halfway down. If you want your audience to visualize themselves looking up (at the base of the cliff) at the nest, then you might use halfway up.
In a slightly different use, imagine a group that is at the top of a cliff and they have a friend climbing up. He reaches the halfway point and one of the friends reports "He's halfway up the cliff." In this case, halfway up implies that he is moving and he is moving up. You can construct a similar case for halfway down.
The given exercise/example is so brief that I don't really see a difference. It's possible that the author was "looking down" or wanted the reader to look down, but I think either choice, down or up, is equally likely and acceptable.
There is no universally accepted list of parts of speech for English. The most traditional list is probably the one you've cited (noun, adjective, pronoun, verb, adverb, preposition, conjunction and interjection), though some would add articles or numerals or both. The term "particle" has no fixed meaning.
Quirk et al. (A Comprehensive Grammar of the English Language) list ten word classes (preposition, pronoun, determiner, conjunction, modal verb, primary verb, full verb, noun, adjective, adverb) plus two "lesser categories" (numerals and interjections) and "a small number of words of unique function" (e.g. the negative particle not and the infinitive marker to).
Huddleston and Pullum (The Cambridge Grammar of the English Language) list nine lexical categories or parts of speech (noun, verb, adjective, adverb, preposition, determinative, subordinator, coordinator, interjection).
So, whereas traditional grammar recognises "some" and "any" as adjectives or pronouns, Quirk et al. classify them as "determiners" (along with articles and numerals), while Huddleston and Pullum regard them as "determinatives". And whereas traditional grammar (and Quirk) recognise "conjunctions", Huddleston and Pullum divide them into "subordinators" and "coordinators" (and also reclassify many words traditionally called conjunctions as prepositions). And while traditional grammar (and Huddleston and Pullum) regard verbs as a single class, Quirk et al. regard them as three separate parts of speech (modal verbs, primary verbs, and full verbs). Finally, whereas traditionalists distinguish pronouns from nouns, H&P regard pronouns as a subclass of nouns (although some traditional pronouns are classed as determinatives).
To answer your questions:
1) words a, an, the (all people know very well they are articles);
They are indeed articles, but this doesn't necessarily mean they are a separate part of speech. The ancient Greeks in fact considered them as such, but many traditional grammarians (following the Romans, whose language lacked articles) have simply treated them as a subclass of adjectives. To this day, some dictionaries put "adj." next to these words.
Modern scholarship regards the articles as a subclass of determiners/determinatives.
2) words like "one", "two", "seven", "eighty" etc. (I consider them numerals);
They are numerals, but that does not necessarily mean they are a separate part of speech - though Quirk et al. specify them as a "minor" word class. Some traditionalists would regard them as adjectives and pronouns. Many people would regard them as a subclass of determiners/determinatives.
3) words like "even" (even John likes apples), "only" (only you can do this job), "just". Are they particles? If not, what part of speech are they?
The OED offers two definitions of "particles": one is that of a minor function word that can't be inflected, and the other is that of adverbs or prepositions used as part of a phrasal verb. By the first definition, they could be considered particles, but most English grammars don't specify particles as a distinct word class. In the dictionaries, "even", "only", "just" are adjectives and adverbs (the latter in your examples).
Best Answer
The first example is ungrammatical. The second example is correct.
The third example is grammatical, but it is not comparable to the second.
The second example "sincerity and courtesy are ... traits", equates the predicate to the subject.
The third example doesn't equate the predicate to the subject, but says that the subject has a particular property.
This corresponds to a particular meaning of "of":
M-W "of" 10
"—used as a function word to indicate a characteristic or distinctive quality or possession, for example, a woman of courage."