Learn Hardcore German: Ich behalte ihr Geheimnis für mich, weil Ehrlichkeit und Vertrauen wichtig für unsere Freundschaft sind. - I keep her secret to myself because honesty and trust are important for our friendship.

Geheimnis is a neuter noun: das Geheimnis.
In the sentence, ihr Geheimnis is a direct object (after behalten), so it’s in the accusative singular neuter.

For possessive determiners like ihr- (her/their):

Neuter accusative singular → ihr Geheimnis (no extra ending)
Feminine accusative singular → ihre Freundin
Masculine accusative singular → ihren Freund

So ihr Geheimnis = her secret (or their secret, depending on context).

Ihres Geheimnisses would be a genitive form, e.g. wegen ihres Geheimnisses (because of her secret), which is not what’s needed here.

How do I know whether ihr here means her or their?

Grammatically, ihr can mean both:

her (3rd person singular feminine)
their (3rd person plural)

In ihr Geheimnis, both readings are possible:

her secret
their secret

German does not mark this difference in this form; you must get it from context (who was being talked about in the previous sentences). This ambiguity is normal in German.

Why do we say für mich behalten? Could I also say bei mir behalten?

The standard idiom is:

etwas für sich behalten = to keep something to oneself

So:

Ich behalte ihr Geheimnis für mich.
= I keep her secret to myself / I don’t tell anyone.

Bei mir behalten is not idiomatic in this sense. You could sometimes hear things like bei mir behalten, but it usually has a more literal meaning like keep it with me (physically), not I won’t tell anyone.

If you just want the idea I won’t reveal it, other natural options are:

Ich halte ihr Geheimnis geheim.
Ich verrate ihr Geheimnis nicht.

What’s the difference between ich behalte ihr Geheimnis für mich and ich halte ihr Geheimnis geheim?

Both express the idea of keeping the secret, but with slightly different focuses:

Ich behalte ihr Geheimnis für mich.
Literally: I keep her secret for myself.
→ Emphasizes not sharing it with others (keeping it to yourself).
Ich halte ihr Geheimnis geheim.
Literally: I keep her secret secret.
→ Emphasizes the state of secrecy and your active choice to maintain it.

In everyday conversation, they’re very close in meaning and both are natural.

Why is the verb at the end in weil Ehrlichkeit und Vertrauen wichtig für unsere Freundschaft sind?

Because of weil. In German:

Weil introduces a subordinate clause (a dependent clause that gives a reason).
In subordinate clauses, the finite verb goes to the end.

So:

Main clause: Ehrlichkeit und Vertrauen sind wichtig für unsere Freundschaft.
With weil: …, weil Ehrlichkeit und Vertrauen wichtig für unsere Freundschaft sind.

If you used denn instead of weil, you would keep normal word order:

…, denn Ehrlichkeit und Vertrauen sind wichtig für unsere Freundschaft.
(Here denn is a coordinating conjunction, so no verb-final order.)

Why is it wichtig für unsere Freundschaft sind and not sind wichtig für unsere Freundschaft?

In a weil-clause, only the finite verb has to go to the final position. The order of the other elements is flexible:

…, weil Ehrlichkeit und Vertrauen wichtig für unsere Freundschaft sind.
…, weil Ehrlichkeit und Vertrauen für unsere Freundschaft wichtig sind.

Both are correct. What must be at the end is sind. Everything else (subject, predicate adjective wichtig, prepositional phrase für unsere Freundschaft) can be arranged before sind for style, rhythm, or emphasis.

Why is it sind (plural) and not ist, since these are abstract nouns?

The subject is a compound subject:

Ehrlichkeit und Vertrauen (two things)

In German, when you have X und Y as subject, the verb is plural, regardless of whether the nouns are abstract or concrete:

Ehrlichkeit und Vertrauen sind wichtig.
Äpfel und Birnen sind teuer.

So sind is required because there are two subjects joined by und.

Why is it für unsere Freundschaft and not für unserer Freundschaft?

Freundschaft is feminine: die Freundschaft.
Für always takes the accusative.
Feminine accusative singular with the possessive unser- is unsere.

So:

Nominative: unsere Freundschaft (our friendship)
Accusative: für unsere Freundschaft (for our friendship)

Unserer Freundschaft would be dative or genitive feminine, used in other structures, e.g.:

mit unserer Freundschaft (with our friendship – dative)
wegen unserer Freundschaft (because of our friendship – genitive)

Why do we use wichtig für + accusative? Could I say wichtig unserer Freundschaft?

The pattern in this sentence is:

wichtig für + Akkusativ
→ wichtig für unsere Freundschaft

This means important for something/someone in the sense of beneficial, relevant, significant to.

You cannot say wichtig unserer Freundschaft here; that’s ungrammatical.

There is another common pattern with wichtig and the dative:

Mir ist Ehrlichkeit wichtig.
(Literally: To me, honesty is important. / Honesty is important to me.)

So:

wichtig für + Akk.: important for X
X ist mir wichtig.: X is important to me (dative of the person)

Could I use da instead of weil in this sentence, and what would be the difference?

Yes, you could say:

Ich behalte ihr Geheimnis für mich, da Ehrlichkeit und Vertrauen wichtig für unsere Freundschaft sind.

Differences:

weil
- Neutral, very common in spoken and written German.
- Focuses on the reason (because).
da
- A bit more formal/literary and often used when the reason is assumed to be already known or is background information.
- Often translated as since or as.

Grammatically, both introduce a subordinate clause, so the verb (sind) still goes to the end.