sigoto de ha sekinin ga daizi desu.

では is で + は.

• で marks the setting, context, or field
• は marks that part as the topic and can add a slight sense of contrast or as for

So 仕事では means something like:

• at work
• in work
• when it comes to work
• in the context of work

It is not just a simple location marker here. It sets up the area in which the statement is true: in work, responsibility is important.

Because で can mark the place or situation where something applies or happens.

In this sentence, 仕事で does not mean that someone is physically doing an action at work in the same way as in a verb sentence like 仕事で食べる. Instead, it gives the context: in the workplace / in one’s job / in the sphere of work.

Compare:

• 学校では礼儀が大事です。
  At school, manners are important.
• 日本では電車が便利です。
  In Japan, trains are convenient.

So 仕事で is helping define the area where 責任が大事 is being said.

Because 責任 is the thing being described as important.

With adjectives like 大事, the noun being evaluated is often marked by が:

• 健康が大事です。
  Health is important.
• 時間が大事です。
  Time is important.
• 責任が大事です。
  Responsibility is important.

So the basic structure is:

• 責任 = responsibility
• が = marks what is being identified/described
• 大事です = is important

Literally, it is close to Responsibility is important.

This is a very common Japanese pattern.

The sentence has:

• a topic: 仕事では = as for work / in work
• a subject inside that topic: 責任が = responsibility

So the structure is basically:

• As for work, responsibility is important.

Japanese often works like this:

• Xでは Yが 大事です。
• In X, Y is important.

More examples:

• 日本では時間が大事です。
  In Japan, time is important.
• スポーツでは練習が大事です。
  In sports, practice is important.

So は and が are not competing here; they are doing different jobs.

Yes, it can be correct, but the nuance changes.

• 責任が大事です simply presents responsibility as the thing that is important.
• 責任は大事です puts stronger topic or contrast emphasis on responsibility, as if saying:
  • As for responsibility, it is important
  • Responsibility, at least, is important
  • or contrasting it with something else

In many neutral statements, が sounds more natural after a context phrase like 仕事では.

So:

• 仕事では責任が大事です。 = natural neutral statement
• 仕事では責任は大事です。 = possible, but more contrastive or emphatic

Yes. 大事 is a na-adjective.

Na-adjectives use な only when they directly modify a noun:

• 大事なこと = an important thing
• 大事な仕事 = important work

But when a na-adjective comes at the end of a sentence as the predicate, you do not use な:

• これは大事です。 = This is important.
• 責任が大事です。 = Responsibility is important.

So in your sentence, 大事です is exactly the correct form.

The plain form is 大事だ.

So:

• polite: 責任が大事です。
• plain: 責任が大事だ。

In slightly softer plain style, especially in writing or casual speech, you may also hear:

• 責任が大事。

That drops だ, which is very common in informal Japanese.

Often, yes.

Because は can imply as for... or when it comes to..., it can suggest a contrast such as:

• In work, responsibility is important
  maybe compared with other contexts
• At work, responsibility matters

But the contrast does not have to be strong. Sometimes it is just a natural topic marker.

So 仕事では責任が大事です can simply mean In work, responsibility is important, without any dramatic contrast.

Your sentence is understandable and grammatically fine. But depending on the exact meaning, Japanese speakers might also say:

• 仕事では責任感が大事です。
  A sense of responsibility is important in work.
• 仕事では責任を持つことが大事です。
  It is important to have/take responsibility at work.

The difference is nuance:

• 責任 = responsibility
• 責任感 = sense of responsibility
• 責任を持つこと = the act of having/taking responsibility

So your original sentence works, but in real-life Japanese, one of these alternatives may sound a little more specific depending on what the speaker wants to emphasize.

The structure is:

• 仕事では = in work / at work / when it comes to work
• 責任が = responsibility
• 大事です = is important

So the pattern is:

• [context/topic] では [thing] が [description] です

A very useful model to remember is:

• 学校では日本語が必要です。
• 旅行では計画が大事です。
• 仕事では責任が大事です。

This pattern is common and very useful for saying In X, Y is important / necessary / convenient / difficult, and so on.