Мы часто пишем друг другу после работы и рассказываем, как прошёл день.

Мы means we and is the subject of both verbs: пишем and рассказываем.

Russian often can omit subject pronouns when they are clear from the verb ending, so Часто пишем друг другу... is possible in context. But мы is included here to make the subject explicit, especially in a full standalone sentence.

Пишем is the 1st person plural form of писать (to write).

• я пишу = I write / am writing
• ты пишешь = you write
• мы пишем = we write

So мы часто пишем means we often write or we often text/message, depending on context.

Because this sentence describes a habitual, repeated action: we often write to each other after work.

• писать is imperfective: ongoing, repeated, habitual action
• написать is perfective: a completed result, like to write up / to write and finish

Since часто (often) suggests repetition, the imperfective пишем is the natural choice.

Друг другу is a fixed expression meaning to each other.

Grammatically, it uses dative case, because писать кому? = to write to whom?

• друг by itself normally means friend
• but in друг другу, it becomes part of an idiomatic reciprocal phrase: one to another / each other

You should learn друг другу as a set phrase.

Examples:

• Мы помогаем друг другу. = We help each other.
• Они звонят друг другу. = They call each other.

Because после requires the genitive case.

• работа = nominative
• работы = genitive

So:

• после работы = after work

This is a common pattern:

• после урока = after the lesson
• после фильма = after the film
• после обеда = after lunch

Usually it means after work in the everyday sense: after finishing the workday.

Russian often uses singular nouns in this kind of expression:

• после работы = after work
• до работы = before work

It does not usually mean after the act of working in some abstract sense. In normal conversation, it is simply the standard way to say after work.

Because Russian often uses рассказывать directly with a clause introduced by как, что, где, etc.

Here:

• рассказываем, как прошёл день literally means something like
• we tell how the day went

In natural English, we often say tell each other about how the day went, but Russian does not need a separate word matching about here.

Because как прошёл день is a subordinate clause.

Main clause:

• Мы часто пишем друг другу после работы и рассказываем...

Subordinate clause:

• как прошёл день

Russian normally separates this kind of clause with a comma.

Because the sentence mixes:

• a habitual present action: we often write / we tell
• with the content of what is being told: how the day went

So the logic is:

• We often write to each other after work and tell how the day went.

The writing/telling happens regularly, but the day is already over by that point, so прошёл is past tense.

Because прошёл agrees with день, which is a masculine singular noun.

• день = masculine
• so the past-tense verb is masculine: прошёл

Compare:

• Как прошла неделя? = How did the week go? (неделя is feminine)
• Как прошло утро? = How did the morning go? (утро is neuter)

Because день is the subject of прошёл.

In the clause как прошёл день, the literal structure is:

• how the day went

So:

• день = subject
• прошёл = verb

That is why день stays in the nominative.

Both are possible, but как прошёл день is the more neutral and common order here.

Russian word order is flexible, and changing it can affect emphasis:

• как прошёл день = neutral
• как день прошёл = more conversational or slightly more emphatic/stylistic

For learners, как прошёл день is the best standard pattern to remember.

Часто means often and shows that the actions are habitual.

It modifies both verbs naturally:

• мы часто пишем... и рассказываем...

So the idea is not a one-time event, but something they do regularly.

In modern Russian, писать can mean literal writing, but in context it often also means to text, to message, or to write to someone online.

So Мы часто пишем друг другу после работы could mean:

• we often write to each other
• we often text each other
• we often message each other

The exact English choice depends on context.

The main stresses are:

• Мы ча́сто пи́шем друг дру́гу по́сле рабо́ты и расска́зываем, как прошёл день.

A few useful notes:

• ча́сто = stress on ча
• пи́шем = stress on пи
• дру́гу = stress on the first syllable
• рабо́ты = stress on бо
• расска́зываем = stress on ска́
• прошёл = stress on шёл

Yes. Russian allows a lot of flexibility.

Possible variants:

• Часто пишем друг другу после работы и рассказываем, как прошёл день.
• После работы мы часто пишем друг другу и рассказываем, как прошёл день.

These all mean basically the same thing, but the emphasis shifts slightly:

• starting with после работы emphasizes when
• including мы makes the subject explicit
• leaving out мы sounds more natural in some conversational contexts

The original version is very neutral and clear for learners.