Der Dynamo und das Rücklicht funktionieren.

Because German nouns have grammatical gender, and these two nouns belong to different genders:

• der Dynamo = masculine
• das Rücklicht = neuter

In this sentence, both nouns are the subjects, so they appear in the nominative case. That is why you see der and das.

German article choice depends on both:

• the noun’s gender
• the noun’s case

So here:

• der = masculine nominative singular
• das = neuter nominative singular

Because the subject is plural overall:

• der Dynamo
• und
• das Rücklicht

Two singular nouns joined by und usually form a plural subject, so the verb must be in the 3rd person plural:

• funktionieren = work / are working

Compare:

• Der Dynamo funktioniert. = The dynamo works.
• Das Rücklicht funktioniert. = The rear light works.
• Der Dynamo und das Rücklicht funktionieren. = The dynamo and the rear light work.

In everyday German, der Dynamo usually refers to a bicycle dynamo or some kind of small generator that produces electricity, especially for bike lights.

So in this sentence, it is probably talking about bicycle equipment:

• der Dynamo = the dynamo / generator
• das Rücklicht = the rear light / tail light

Rücklicht is a compound noun:

• Rück- = rear / back
• Licht = light

So Rücklicht literally means rear light or tail light.

In German compound nouns, the grammatical gender usually comes from the last part of the compound. Here the last part is Licht, and das Licht is neuter. That is why the full compound is also neuter:

• das Licht
• das Rücklicht

In German, all nouns are capitalized. That is a standard spelling rule.

So in this sentence:

• Dynamo is capitalized because it is a noun
• Rücklicht is capitalized because it is a noun

But:

• der, und, das, funktionieren are not nouns, so they are not capitalized

It is the present tense, and in German the present tense often covers both ideas:

• work
• are working

So Der Dynamo und das Rücklicht funktionieren can mean:

• The dynamo and the rear light work.
• The dynamo and the rear light are working.

German does not usually need a separate progressive form like English are working.

It is not actually at the end because of a special rule like in subordinate clauses. It is in the normal main-clause second-position pattern.

The first element is the whole subject:

• Der Dynamo und das Rücklicht

Then comes the finite verb:

• funktionieren

So the structure is:

• [Subject] [Verb]

This is completely normal German word order for a statement.

Because German normally does not use a comma before und when it simply connects two equal elements.

Here und joins two nouns:

• der Dynamo
• das Rücklicht

So no comma is needed.

Because each noun keeps its own singular article:

• der Dynamo
• das Rücklicht

The plural meaning comes from the fact that the two nouns are joined by und. German does not replace them with one shared plural article here.

So the noun phrases remain singular individually, but together they form a plural subject, which is why the verb is plural.

They are in the nominative case because they are the subject of the sentence — they are the things doing the action of functioning.

That is why the articles are:

• der for masculine nominative singular
• das for neuter nominative singular

If the nouns were in a different role, the articles might change. For example:

• Ich sehe den Dynamo. = I see the dynamo.
• Ich sehe das Rücklicht. = I see the rear light.

A rough guide is:

• Rück sounds somewhat like ruek, but with the German ü, which has no exact English equivalent
• licht sounds like liht, with the German ch sound, not a hard English k

A few pronunciation points:

• ü is a front rounded vowel; English speakers often need practice with it
• ck gives a short preceding vowel, so Rück is short
• the ch in Licht is the soft sound heard in words like ich

A rough learner-friendly approximation is:

• RUEKH-likht

Not perfect, but useful as a starting point.

Normally, no. With two singular nouns joined by und, standard German uses a plural verb:

• Der Dynamo und das Rücklicht funktionieren.

Using funktioniert here would usually sound ungrammatical.

The singular verb is used when the subject is singular, for example:

• Der Dynamo funktioniert.
• Das Rücklicht funktioniert.

Yes. In a yes/no question, German usually puts the finite verb first:

• Funktionieren der Dynamo und das Rücklicht?

That means:

• Are the dynamo and the rear light working?

So the statement is:

• Der Dynamo und das Rücklicht funktionieren.

And the yes/no question is:

• Funktionieren der Dynamo und das Rücklicht?

Yes, very common. It is used a lot for things that work, function, or operate.

Examples:

• Der Computer funktioniert. = The computer works.
• Die Lampe funktioniert nicht. = The lamp doesn’t work.
• Alles funktioniert. = Everything works.

So in your sentence, funktionieren is a very natural verb choice for equipment like a dynamo and a rear light.