Obwohl die Bibliothek geschlossen ist, funktioniert der Rückgabeautomat.

Obwohl means although or even though. It introduces a contrast:

• Obwohl die Bibliothek geschlossen ist = Although the library is closed
• funktioniert der Rückgabeautomat = the return machine works / is working

So the sentence expresses that one thing is true despite another thing.

Because obwohl introduces a subordinate clause. In German, subordinate clauses usually send the conjugated verb to the end.

So:

• die Bibliothek ist geschlossen = the library is closed
  (normal main clause word order)
• obwohl die Bibliothek geschlossen ist = although the library is closed
  (subordinate clause, so ist goes to the end)

This is one of the most important word-order patterns in German.

Because the sentence begins with the subordinate clause:

• Obwohl die Bibliothek geschlossen ist, ...

In German main clauses, the conjugated verb must be in second position. The entire subordinate clause counts as the first element, so the verb of the main clause comes immediately after it:

• Obwohl die Bibliothek geschlossen ist, funktioniert der Rückgabeautomat.

If you start with the main clause instead, you get:

• Der Rückgabeautomat funktioniert, obwohl die Bibliothek geschlossen ist.

Both are correct. The word order changes because German is following the verb-second rule.

Yes. The comma is required because obwohl die Bibliothek geschlossen ist is a subordinate clause, and German separates subordinate clauses from main clauses with a comma.

So this is correct:

• Obwohl die Bibliothek geschlossen ist, funktioniert der Rückgabeautomat.

Leaving out the comma would be incorrect in standard written German.

Because der Rückgabeautomat is the subject of the main clause, so it is in the nominative case.

In:

• funktioniert der Rückgabeautomat

the machine is the thing that functions, so it is the subject.

• der = nominative masculine
• den = accusative masculine

Since nobody is acting on the machine here, but the machine itself is doing the action, nominative is needed.

Rückgabeautomat is a compound noun:

• Rückgabe = return
• Automat = machine / automatic machine

So Rückgabeautomat means something like:

• return machine
• book return machine
• automated return machine

In a library context, it is the machine where you return books or other borrowed items.

In German compound nouns, the last part determines the gender.

Here the last part is:

• Automat = masculine

So the whole compound is masculine:

• der Automat
• der Rückgabeautomat

This is a very useful rule for German compounds.

It is the past participle of schließen (to close), but here it is being used in a way that works like an adjective:

• Die Bibliothek ist geschlossen. = The library is closed.

This is a very common pattern in German:

• Die Tür ist geöffnet. = The door is open.
• Das Geschäft ist geschlossen. = The shop is closed.

So in this sentence, geschlossen describes the state of the library.

Yes, in everyday spoken German, zu is often used instead of geschlossen:

• Die Bibliothek ist zu. = The library is closed.

So you could say:

• Obwohl die Bibliothek zu ist, funktioniert der Rückgabeautomat.

That sounds more informal and conversational.
Geschlossen is a bit more formal or neutral and is very common in writing and signs.

The basic main-clause order is:

• Die Bibliothek = subject
• ist = conjugated verb
• geschlossen = predicate adjective / participle

So:

• Die Bibliothek ist geschlossen.

When this becomes a subordinate clause after obwohl, the verb moves to the end:

• obwohl die Bibliothek geschlossen ist

That change is purely about clause type, not meaning.

Yes. German allows both orders:

• Obwohl die Bibliothek geschlossen ist, funktioniert der Rückgabeautomat.
• Der Rückgabeautomat funktioniert, obwohl die Bibliothek geschlossen ist.

The meaning is basically the same. Starting with obwohl puts the contrast up front and often sounds a bit more emphatic.

They are related, but they are used differently.

Obwohl is a subordinating conjunction, so it introduces a subordinate clause and sends the verb to the end:

• Obwohl die Bibliothek geschlossen ist, funktioniert der Rückgabeautomat.

Trotzdem is an adverb, not a conjunction. It usually introduces a new main clause:

• Die Bibliothek ist geschlossen. Trotzdem funktioniert der Rückgabeautomat.

So:

• obwohl = although / even though
• trotzdem = nevertheless / even so / still

They express a similar contrast, but the grammar is different.

It can mean either, depending on context. German present tense often covers both the simple present and the present progressive of English.

So:

• Der Rückgabeautomat funktioniert.

can mean:

• The return machine works.
• The return machine is working.

In this sentence, the idea is probably that the machine is operational even though the library building is closed.