Die Funktionsprüfung gelingt.

Because Funktionsprüfung is a feminine noun, so its nominative singular article is die.

A useful rule for compound nouns in German:

• the last part determines the gender

Here, the last part is Prüfung (test, inspection, check), and Prüfung is feminine:

• die Prüfung

So the whole compound is:

• die Funktionsprüfung

Funktionsprüfung is a compound noun made up of:

• Funktion = function
• Prüfung = test, inspection, check

So Funktionsprüfung means something like:

• function test
• functional test
• operational check

German very often builds technical vocabulary this way by combining nouns into one longer word.

Because German normally writes compound nouns as one word.

So where English often uses:

• function test
• safety check
• quality control

German often uses:

• Funktionsprüfung
• Sicherheitsprüfung
• Qualitätskontrolle

That is completely normal and very common, especially in technical or formal language.

That -s- is a linking element used in many German compounds.

So:

• Funktion
  • Prüfung becomes Funktionsprüfung

This -s- does not usually have a separate meaning by itself. It is just a common way German connects parts of a compound. You will see the same thing in many words, for example:

• Arbeitszimmer
• Lebensmittel
• Geburtsdatum

Unfortunately, there is not always a simple rule for when the linking -s- appears, so learners usually pick it up word by word.

Because gelingt is the 3rd person singular form, and the subject die Funktionsprüfung is singular.

The verb is gelingen. Its present tense forms include:

• ich gelinge
• du gelingst
• er/sie/es gelingt
• wir gelingen
• ihr gelingt
• sie/Sie gelingen

Since die Funktionsprüfung is a singular noun, German uses:

• gelingt

Gelingen means something like:

• to succeed
• to turn out well
• to be successful

So Die Funktionsprüfung gelingt means that the function test is successful or succeeds.

A very important point: gelingen does not mean to succeed at doing something in the same way English often uses succeed with a person as subject. In German, the thing itself is often the subject.

For example:

• Der Versuch gelingt. = The attempt succeeds.
• Die Reparatur gelingt. = The repair is successful.

Because the idea of success is already built into the verb gelingen.

So German does not need a separate adverb here.
Compare:

• Die Funktionsprüfung gelingt. = The function test succeeds / is successful.

If you added something like erfolgreich, it would usually change the structure:

• Die Funktionsprüfung ist erfolgreich.

That is also possible, but it is a different construction.

It is in the nominative case, because it is the subject of the sentence.

You can see this from the sentence structure:

• Die Funktionsprüfung = the thing performing the grammatical role of subject
• gelingt = the verb

So:

• Die Funktionsprüfung gelingt. literally works like:
• The function test succeeds.

There is no direct object here.

No, gelingen is normally intransitive, so it does not take a direct object.

Very often it appears with:

• a subject: the thing that succeeds
• sometimes a dative person: the person for whom it succeeds

For example:

• Die Funktionsprüfung gelingt dem Techniker. = The function test is successful for the technician / The technician succeeds with the function test.

In your sentence, the dative person is simply not mentioned:

• Die Funktionsprüfung gelingt.

They are related in meaning, but they are not the same.

• gelingen = to succeed, to turn out well
• funktionieren = to function, to work

Compare:

• Die Funktionsprüfung gelingt. = The test is successful.

• Das Gerät funktioniert. = The device works.

If you say:

• Die Funktionsprüfung funktioniert it sounds less natural in many contexts, because a test does not usually function in the same way a machine does. A test more naturally succeeds, fails, is successful, or is carried out.

Yes. That is a possible alternative.

Compare:

• Die Funktionsprüfung gelingt.

  • slightly more verbal and event-focused
  • the test succeeds

• Die Funktionsprüfung ist erfolgreich.

  • more descriptive
  • the test is successful

Both can be correct, though the preferred wording may depend on context and style.

Yes, but that means something slightly different in tense.

• Die Funktionsprüfung gelingt. = present tense
  = The function test succeeds / is succeeding

• Die Funktionsprüfung ist gelungen. = perfect tense of gelingen
  = The function test has succeeded / was successful

So ist gelungen is usually used when the successful result is already complete.

Because German main clauses normally follow the verb-second rule.

In this sentence:

• Die Funktionsprüfung = first element
• gelingt = second element

So the normal word order is:

• Die Funktionsprüfung gelingt.

You could also move something else to the first position, but the verb would still stay second:

• Heute gelingt die Funktionsprüfung.

Yes. Funktionsprüfung is definitely a technical/formal word, and the whole sentence sounds like something you might see in:

• engineering documentation
• testing reports
• manuals
• quality assurance contexts

In everyday conversation, people might use simpler wording depending on context.

The main stress is usually on the first part of the compound:

• FUNK-tions-prü-fung

A rough English approximation is:

• FOONK-tsi-ons-proo-fung

A few pronunciation notes:

• ü is a front rounded vowel, not exactly like English oo
• prüfung has the ü sound again
• the tion in Funktion is pronounced roughly like tsyohn

If pronunciation matters for you, it is helpful to practice:

• Funktion
• Prüfung
• Funktionsprüfung

separately first, then together.

In German, all nouns are capitalized, but verbs are not unless they start the sentence.

So here:

• Die = article
• Funktionsprüfung = noun, so capitalized
• gelingt = verb, so lowercase

This is one of the most noticeable spelling features of German.