Odanın karanlık olmasına rağmen gizemi çözdük.

It has two parts:

• Odanın karanlık olmasına rağmen = despite the room being dark / although the room was dark
• gizemi çözdük = we solved the mystery

So the sentence is built as:

[despite X] + [main action]

A fairly literal translation would be:

Despite the room’s being dark, we solved the mystery.

Because Turkish often leaves subject pronouns out when the verb already shows the person.

Here, çözdük already means we solved:

• çöz- = solve
• -dü- = past tense
• -k = we

So biz is not necessary unless you want extra emphasis.

Oda means room.
Odanın means of the room / the room’s.

In this sentence, the part odanın karanlık olması acts like a noun-like clause meaning:

the room’s being dark

That is why oda appears in the genitive form odanın. It marks the subject of the embedded, nominalized clause.

Olmasına can be broken down like this:

• ol- = be / become
• -ma = verbal noun marker
• -sı = 3rd person singular possessive marker
• -na = dative case

So olması means something like its being or the fact that it is.
Then olmasına is the dative form, because rağmen normally follows something in the dative.

So:

karanlık olmasına rağmen
literally = despite its being dark

Because karanlık means dark, but the sentence needs the idea of being dark.

Turkish commonly uses olmak to build that kind of clause:

• karanlık = dark
• karanlık olmak = to be dark
• karanlık olması = being dark
• karanlık olmasına rağmen = despite being dark

So olmak is what lets the adjective become part of a full clause-like structure.

Rağmen means despite / in spite of.

It is a postposition, not a preposition. That means it comes after the word or clause it belongs to.

Also, it usually follows a dative form:

• buna rağmen = despite this
• yağmura rağmen = despite the rain
• olmasına rağmen = despite being / although it was

So in your sentence, rağmen connects to olmasına.

Because gizemi is the definite accusative form.

• gizem = mystery
• gizemi = the mystery

Turkish often marks a direct object with the accusative when it is specific or definite. Here, the sentence is talking about a particular mystery, so gizemi is natural.

Çözdük breaks down as:

• çöz- = solve
• -dü- = past tense
• -k = we

So çözdük means we solved.

Because of vowel harmony, the past tense appears here as -dü rather than another form like -dı or -du.

No, Turkish word order is flexible, but this version is very natural.

The neutral pattern here is:

background/circumstance + object + verb

So:

Odanın karanlık olmasına rağmen gizemi çözdük.

You could move parts around for emphasis, such as:

Gizemi, odanın karanlık olmasına rağmen çözdük.

But the original sentence sounds normal and straightforward.

Yes. A very common alternative is:

Oda karanlık olsa da gizemi çözdük.

This also means Although the room was dark, we solved the mystery.

Compared with olmasına rağmen, olsa da can sound a bit lighter and more conversational.
Your original sentence is perfectly good, and may sound a little more formal or emphatic.