Честность важна в дружбе и на работе.

Because важна is the short form of the adjective важный (important), and short-form adjectives are commonly used in Russian when the adjective is the main predicate of the sentence.

So:

• Честность важна. = Honesty is important.

By contrast, the long form важная is usually used before a noun:

• важная работа = important work
• важная тема = an important topic

So in this sentence, важна is the natural choice.

In Russian, the verb to be is normally omitted in the present tense.

So:

• Честность важна literally looks like Honesty important
• but it means Honesty is important

This is completely normal Russian grammar. If you want past or future, then a form of быть appears:

• Честность была важна. = Honesty was important.
• Честность будет важна. = Honesty will be important.

Because it agrees with честность, which is a feminine singular noun.

• честность = feminine
• therefore the short-form adjective must also be feminine:
  • masculine: важен
  • feminine: важна
  • neuter: важно
  • plural: важны

Examples:

• Человек важен. = The person is important.
• Честность важна. = Honesty is important.
• Решение важно. = The decision is important.
• Принципы важны. = Principles are important.

Both дружбе and работе are in the prepositional case.

That is because they come after the prepositions:

• в in в дружбе
• на in на работе

Here those prepositions express a setting, sphere, or context, so they take the prepositional case.

Base forms:

• дружба → в дружбе
• работа → на работе

So the sentence means something like:

• in friendship and at work
• or more naturally, in friendships and at work

Because Russian uses different prepositions idiomatically with different nouns.

• в дружбе literally means in friendship
• на работе means at work

For English speakers, this is mostly something to learn as a collocation.

A useful way to think of it:

• в often means in / within
• на often means on / at, but in many expressions it simply has to be memorized

For example:

• в школе = at school / in school
• на работе = at work
• в жизни = in life
• на практике = in practice

So на работе is just the normal Russian way to say at work.

Usually it means in friendship in a general, abstract sense.

Russian has no articles like a or the, so context does that work. In this sentence, the meaning is broad and general:

• Honesty is important in friendship and at work.

It is talking about friendship as a sphere of life, not one specific friendship.

Yes. Russian word order is more flexible than English word order, although the basic sentence here is already very natural.

Original:

• Честность важна в дружбе и на работе.

Possible variations:

• В дружбе и на работе честность важна.
• На работе и в дружбе честность важна.

These all mean roughly the same thing, but the emphasis changes slightly.

• Starting with Честность puts the focus first on honesty
• Starting with В дружбе и на работе puts the context first: in friendship and at work

Yes, честность is the normal noun meaning honesty.

It is formed from the adjective:

• честный = honest

and the noun-forming suffix:

• -ость

So:

• честный → честность
• honest → honesty

This pattern is very common in Russian:

• серьёзный → серьёзность = serious → seriousness
• нежный → нежность = tender → tenderness
• вежливый → вежливость = polite → politeness

The main stress is:

• ЧЕстность важНА в дружБЕ и на раБОте.

A rough pronunciation guide for an English speaker:

• CHES-nast' vazh-NA v droozh-BYE ee na ra-BO-tye

A few notes:

• ч sounds like ch in check
• ж sounds like the s in measure
• unstressed о is often pronounced more like a
• the final -е in дружбе and работе is pronounced roughly like ye

Yes, but that would be a slightly different structure.

• Честность важна. = Honesty is important.
• Честность — это важно. = Honesty—that is important.

The version with это can sound a bit more emphatic, explanatory, or general. But for your sentence, Честность важна в дружбе и на работе is the cleaner and more standard phrasing.

Because Russian often uses the singular to talk about something in a general sense.

So:

• в дружбе = in friendship
• not necessarily in one friendship, but in the whole concept or sphere of friendship

English does this too sometimes:

• Honesty is important in life
• Patience matters in education

Russian uses the singular very naturally in this kind of abstract statement.

A little, yes.

• Честность важна в дружбе и на работе. sounds like a general statement or principle
• Важно быть честным в дружбе и на работе. = It is important to be honest in friendship and at work

Both are correct and natural, but they focus slightly differently:

• честность emphasizes the quality itself: honesty
• быть честным emphasizes the action or behavior: to be honest

Both are useful, and Russian uses both kinds of phrasing often.