Umiejętność spokojnej rozmowy z klientem jest ważna.

Because umiejętność often takes a noun in the genitive case.

• rozmowa = nominative singular, conversation
• rozmowy = genitive singular, of conversation

So umiejętność rozmowy literally means the skill/ability of conversation, which English would usually express as the ability to talk / to have a conversation.

In this sentence:

• umiejętność = the skill
• spokojnej rozmowy z klientem = of calm conversation with a client

Because spokojnej is an adjective, and it describes the noun rozmowy.

• spokojna rozmowa = a calm conversation
• spokojnej rozmowy = of a calm conversation

If you used spokojnie, that would be an adverb, meaning calmly, and it would describe how someone speaks, not the noun conversation itself.

So:

• spokojnej rozmowy = calm conversation
• spokojnie rozmawiać = to talk calmly

Both ideas are possible in Polish, but this sentence is built with a noun phrase, so the adjective form spokojnej is needed.

Because it has to agree with rozmowy in gender, number, and case.

Here rozmowy is:

• feminine
• singular
• genitive

So the adjective also becomes:

• feminine
• singular
• genitive

That gives spokojnej.

Compare:

• spokojna rozmowa = nominative
• spokojnej rozmowy = genitive

Because the preposition z meaning with takes the instrumental case.

So:

• klient = client
• klientem = with a client

This is a very common Polish pattern:

• z nauczycielem = with a teacher
• z kobietą = with a woman
• z dzieckiem = with a child

Important: z does not always take the same case.

• z + instrumental = with
• z + genitive can mean from / off / out of

Here it clearly means with a client, so klientem is instrumental.

Because ważna agrees with the subject umiejętność, and umiejętność is a feminine singular noun.

So:

• ważny = masculine singular
• ważna = feminine singular
• ważne = neuter singular, or some plural uses

Since the subject is umiejętność, we say:

• Umiejętność ... jest ważna.

A useful thing to remember: many nouns ending in -ość are feminine, including:

• umiejętność = skill
• możliwość = possibility
• trudność = difficulty

Normally, no.

In a regular present-tense sentence like this, jest is usually needed:

• Umiejętność spokojnej rozmowy z klientem jest ważna.

Leaving it out would sound unnatural in normal speech, though you might see omission in:

• headlines
• notes
• poetic style
• very informal shorthand

So for standard Polish, keep jest.

Not completely. Polish word order is more flexible than English because the endings show the grammatical roles.

The sentence as given is neutral and natural:

• Umiejętność spokojnej rozmowy z klientem jest ważna.

You could also say:

• Ważna jest umiejętność spokojnej rozmowy z klientem.

That puts more emphasis on ważna.

So the basic meaning stays the same, but word order can change the focus or emphasis.

Because Polish has no articles.

English must choose between a client and the client, but Polish just says:

• klient
• klientem
• etc.

So z klientem can mean:

• with a client
• with the client

The exact meaning comes from context.

The same is true for umiejętność:

• a skill
• the skill

Polish leaves that unstated unless the context makes it clear.

Yes. Polish often has several natural ways to express the same basic idea.

For example:

• Umiejętność spokojnego rozmawiania z klientem jest ważna.
  = the skill of talking calmly with a client is important

• Spokojne rozmawianie z klientem jest ważne.
  = talking calmly with a client is important

• Spokojna rozmowa z klientem jest ważna.
  = a calm conversation with a client is important

These are similar, but not identical:

• Umiejętność ... jest ważna focuses on the skill/ability
• Rozmawianie ... jest ważne focuses on the activity
• Rozmowa ... jest ważna focuses on the conversation itself

So the original sentence specifically emphasizes that the ability is important.