Der Fehlercode zeigt ein Problem mit dem Kältemittel im Kältekreis.

Because Fehlercode is a masculine noun in German, so its dictionary form is der Fehlercode.

A useful rule here is that in German, the last part of a compound noun usually determines the gender.

• der Code = the code
• der Fehlercode = error code

So even though Fehler is part of the word, the gender comes from Code, which is masculine.

Because the subject is der Fehlercode, which is third person singular: the error code.

German verbs change depending on the subject:

• ich zeige
• du zeigst
• er/sie/es zeigt
• wir zeigen

Since der Fehlercode is like er grammatically, the correct form is zeigt.

Because Problem is a neuter noun: das Problem.

Here it is the direct object of zeigt, so it is in the accusative case. For neuter nouns, the indefinite article stays ein in both nominative and accusative:

• nominative: ein Problem
• accusative: ein Problem

Compare that with a masculine noun:

• nominative: ein Fehler
• accusative: einen Fehler

So ein Problem is correct.

Because the preposition mit always takes the dative case.

So:

• das Kältemittel = the refrigerant/coolant
• dative singular: dem Kältemittel

That gives:

• mit dem Kältemittel = with the refrigerant / involving the refrigerant

This is a very important pattern to remember:

• mit dem for masculine or neuter singular nouns in the dative
• mit der for feminine singular nouns
• mit den for plural nouns

im is simply a contraction of in dem.

So:

• in dem Kältekreis = literally correct
• im Kältekreis = the normal contracted form

German often contracts:

• in dem → im
• an dem → am
• zu dem → zum
• zu der → zur

So im Kältekreis means in the cooling circuit / refrigeration circuit.

Because here in is being used to describe location, not movement.

German in can take:

• accusative for movement into something
• dative for being in something

Compare:

• Ich gehe in den Raum. = I go into the room.
  
  • movement → accusative
• Ich bin im Raum. = I am in the room.
  
  • location → dative

In your sentence, the problem is located in the cooling circuit, so German uses the dative:

• im Kältekreis = in dem Kältekreis

Because German very often forms compound nouns by joining words together into a single noun.

Examples in this sentence:

• Fehler + Code → Fehlercode
• Kälte + Mittel → Kältemittel
• Kälte + Kreis → Kältekreis

This is extremely common in German. English often writes similar ideas as two words or with of, but German tends to compress them into one noun.

Also, all nouns in German are capitalized, so:

• Fehlercode
• Problem
• Kältemittel
• Kältekreis

all begin with capital letters.

That -e- comes from the noun Kälte.

• die Kälte = cold
• das Mittel = means/agent/substance
• der Kreis = circle/circuit

So:

• Kältemittel = literally something like cold-agent/substance
• Kältekreis = cooling/refrigeration circuit

This is just how these compounds are formed. When you learn German compounds, it is often helpful to identify the base words, even if the exact English translation is more technical.

Yes, German could use other verbs depending on context, but zeigt is perfectly natural.

Nuances:

• zeigen = to show
• anzeigen = to indicate/display
• melden = to report/signal

So these are all possible in technical language:

• Der Fehlercode zeigt ein Problem ...
• Der Fehlercode zeigt ein Problem an ...
• Der Fehlercode meldet ein Problem ...

Your sentence sounds natural and straightforward. In manuals or diagnostics, anzeigen is also very common.

The basic structure is:

Der Fehlercode | zeigt | ein Problem | mit dem Kältemittel | im Kältekreis

So:

1. subject = Der Fehlercode
2. verb = zeigt
3. object = ein Problem
4. prepositional phrase explaining the problem = mit dem Kältemittel
5. prepositional phrase giving location/context = im Kältekreis

German main clauses usually place the finite verb in second position, and that is exactly what happens here.

The letter ä is different from plain a.

In these words:

• Fehlercode
• Kältemittel
• Kältekreis

the ä in Kälte- sounds somewhat like the vowel in English care for many learners, though it is not exactly the same.

A rough learner-friendly guide:

• Kälte ≈ KEL-teh
• Kältemittel ≈ KEL-teh-mit-tel
• Kältekreis ≈ KEL-teh-kryce

Also note:

• ie in Problem is not present, so Problem is pronounced roughly pro-BLAYM
• ei in zeigt sounds like English eye

These are only approximations, but they can help at the beginning.

Yes, but the verb still has to stay in second position in a normal main clause.

For example, you could also say:

• Im Kältekreis zeigt der Fehlercode ein Problem mit dem Kältemittel.

That puts more emphasis on im Kältekreis. Notice that when something else comes first, the verb still stays second:

• Im Kältekreis | zeigt | der Fehlercode ...

So German word order is flexible, but the verb-second rule is very important.

Because German is using the indefinite article ein, which means a/an, not the.

So:

• ein Problem = a problem
• das Problem = the problem

In many technical or diagnostic sentences, German often uses the indefinite article because it is identifying the type of issue rather than a specific previously known one.

So the sentence presents a problem, not necessarily the specific problem already known to both speaker and listener.