Encontramos la causa.

Spanish often leaves out subject pronouns because the verb ending already shows who the subject is.

• encontramos can mean we find or we found
• so nosotros is usually unnecessary

You can say Nosotros encontramos la causa, but that normally adds emphasis, contrast, or clarity, for example if you want to stress we rather than someone else.

It can mean either one.

In isolation, Encontramos la causa is ambiguous:

• we find the cause = present tense
• we found the cause = preterite/simple past

This happens because for many -ar verbs, the nosotros form is the same in the present and the preterite.

Context usually makes it clear:

• Ahora encontramos la causa = present
• Ayer encontramos la causa = past

Because encontrar is a stem-changing verb only in certain present-tense forms.

The usual present-tense pattern is o → ue:

• encuentro
• encuentras
• encuentra
• encontramos
• encontráis
• encuentran

In the nosotros and vosotros forms, the stem change does not happen, so it stays encontramos, not encuentramos.

Also, in the preterite, there is no stem change here either:

• encontré, encontraste, encontró, encontramos...

It is mostly regular, but with a present-tense stem change.

So:

• In the present, it behaves like a stem-changing -ar verb: o → ue
• In the preterite, it is regular: encontré, encontraste, encontró, encontramos...

That means encontramos is a perfectly normal form both for:

• present nosotros
• preterite nosotros

La means you are talking about a specific cause, not just any cause.

• la causa = the cause
• una causa = a cause

So Encontramos la causa suggests that the cause is already known, expected, or identifiable in the context. If you said Encontramos una causa, it would mean you found one possible cause, not necessarily the main or already-mentioned one.

Yes. Causa is a feminine noun, so it takes the feminine singular article la.

• la causa
• plural: las causas

For English speakers, this is just a gender feature of the noun. It does not mean the cause is somehow feminine in meaning; it is simply grammatical gender.

Because the personal a is mainly used with specific human beings, and sometimes pets or personified beings, when they are direct objects.

Examples:

• Veo a María
• Encontramos al médico

But la causa is a thing, not a person, so there is no personal a:

• Encontramos la causa

Yes. That is also correct, but it is a different tense.

• Encontramos la causa = we found the cause / we find the cause
• Hemos encontrado la causa = we have found the cause

In Spain, hemos encontrado is very commonly used for a past action connected to the present, especially if it happened recently or within a current time period. So in Peninsular Spanish, both forms are possible, but the choice depends on context and nuance.

Yes, Spanish word order is more flexible than English word order.

The neutral order here is:

• Encontramos la causa

But you may also hear variations for emphasis:

• Nosotros encontramos la causa = emphasizes we
• La causa la encontramos = emphasizes the cause

That said, the original sentence is the most straightforward and neutral version.