Мы так и не смогли сразу выяснить, почему письмо попало в спам, но потом всё равно нашли его.

Because the subject is мы = we, so the verb must agree with it in number.
Смогли is the past plural form of смочь (to manage to, to be able to in a completed situation).

• я смог / смогла = I managed
• ты смог / смогла = you managed
• мы смогли = we managed

So Мы ... не смогли means We didn’t manage to...

Because смочь works like to manage to / to be able to and is followed by an infinitive describing the action that was or wasn’t possible.

So:

• смогли выяснить = managed to find out
• не смогли выяснить = didn’t manage to find out

This is very normal Russian structure:

• Я смог прийти = I managed to come
• Мы не смогли понять = We couldn’t understand

Так и не is a very common Russian combination meaning something like:

• still didn’t
• never did
• ended up not
• failed to, despite expectation

In this sentence, Мы так и не смогли... suggests that there was an attempt or expectation, but the result never happened.

So Мы так и не смогли сразу выяснить... is stronger than just Мы не смогли сразу выяснить.... It carries the feeling of:

We never did manage to immediately figure out...

Examples:

• Я так и не понял = I never did understand
• Они так и не пришли = They never ended up coming

Сразу means right away, immediately, at once.

Here it modifies выяснить:

• не смогли сразу выяснить = couldn’t find out immediately

So the sentence does not mean they never found out anything at all. It means they could not determine it right away.

Выяснить means to выяснить something fully / to determine / to find out / to clarify. It often suggests getting to the bottom of something.

Here it fits because the sentence is about determining the reason:

• выяснить, почему... = find out why...

It is slightly more formal or precise than simply узнать in some contexts.

Compare:

• узнать = to learn/find out a fact
• выяснить = to investigate/clarify/determine

So выяснить, почему письмо попало в спам sounds like figure out why the email went to spam.

Because the sentence refers to a single completed result: either they managed to find out, or they didn’t.

Выяснить is the perfective infinitive, focusing on the result:

• смогли выяснить = managed to find out
• не смогли выяснить = didn’t manage to find out

If you used выяснять, it would sound more like an ongoing process, which is less natural after смочь here.

Because попало в спам means ended up in spam / got into spam, which emphasizes the event of going there.

• попало в спам = it got sent/filtered into the spam folder
• было в спаме = it was in the spam folder

The sentence is about the reason this happened, so the event verb попасть fits better.

Because попасть expresses movement or change of destination: the email went into spam. After в, Russian uses:

• accusative for movement into something
• prepositional for location inside something

So:

• попало в спам = went into spam
• было в спаме = was in spam

This is the same pattern as:

• положить в коробку = put into the box
• лежать в коробке = lie in the box

Grammatically, письмо just means letter/message, but in this context, because of спам, it is naturally understood as an email.

Russian often uses письмо for both:

• a physical letter
• an email

If needed, speakers can clarify with электронное письмо, but usually context is enough.

Because письмо is the subject of the subordinate clause:

почему письмо попало в спам

Here, письмо is the thing that попало. So it is the grammatical subject, and subjects are usually in the nominative case.

Here всё равно means all the same / anyway / nevertheless.

So:

• но потом всё равно нашли его = but later we found it anyway

It shows that even though they couldn’t immediately figure out why it went to spam, the result was still that they found the message.

Common uses:

• Я всё равно пойду = I’ll go anyway
• Он всё равно не согласится = He still won’t agree / He won’t agree anyway

Because in Russian, once the subject is already clear, it is often omitted in the next clause.

So:

• Мы ... не смогли... , но потом всё равно нашли его literally has an implied we in the second part.

Russian does this very naturally when the subject stays the same.

You could say но потом мы всё равно нашли его, but leaving out мы is perfectly normal and often more natural.

Russian word order is flexible. Нашли его is natural here because the speaker is ending the sentence with the object for smooth information flow.

Both are possible:

• нашли его
• его нашли

But they can sound slightly different in emphasis.

In this sentence, нашли его sounds neutral and natural after но потом всё равно.

It refers back to письмо.

Even though письмо is neuter, the pronoun is его, because in the accusative singular for inanimate nouns/pronouns, masculine/neuter forms look the same here.

So:

• письмо = neuter noun
• нашли его = found it

This is the normal pronoun for it in this context.

Because the sentence talks about concrete events and results:

• смогли = managed / did not manage
• выяснить = find out completely
• попало = ended up in spam
• нашли = found

Perfective aspect is very common when narrating completed events in a sequence. The sentence is basically telling a short story:

1. we didn’t manage to find something out right away,
2. the email had gone to spam,
3. later we found it anyway.

So perfective aspect matches that result-focused narration very well.