Получатель письма попросил меня сохранить скриншот и написать короткий запрос администратору.

Because Russian often shows of relationships with the genitive case.

• получатель = recipient
• письма = of the letter

So получатель письма literally means the recipient of the letter.

Here, письма is the genitive singular of письмо.

That is a very common confusion. The form письма can indeed mean different things in different contexts:

• genitive singular: of the letter
• nominative plural: letters

In получатель письма, it must be genitive singular, because получатель is followed by a noun meaning recipient of what? → of the letter.

So here it does not mean recipient of letters in general. It means recipient of the letter.

попросил is the past tense, masculine singular form of попросить.

It tells you two things:

1. The action happened in the past: asked
2. The subject is grammatically masculine singular: получатель

So:

• получатель ... попросил = the recipient ... asked

Russian past tense verbs agree in gender and number with the subject:

• попросил = masculine
• попросила = feminine
• попросило = neuter
• попросили = plural

Because with попросить + someone + infinitive, the person being asked is usually in the accusative case.

So:

• попросил меня сохранить... = asked me to save...

Here:

• я → меня in the accusative

A learner often expects something like to me, but Russian uses the direct-object pattern here.

Compare:

• Он попросил меня помочь. = He asked me to help.

Because after попросил меня, Russian often uses an infinitive to say what the person was asked to do.

So the structure is:

• попросил меня + infinitive
• asked me to + verb

Examples:

• попросил меня подождать = asked me to wait
• попросил меня сохранить = asked me to save
• попросил меня написать = asked me to write

That is why both actions appear as infinitives.

Because the sentence is talking about completed, one-time tasks.

• сохранить = to save as a completed action
• написать = to write as a completed action

After попросил, perfective infinitives are very natural when someone asks for a concrete result:

• save the screenshot
• write the request

If you used the imperfective forms, the meaning would shift more toward process, repetition, or general activity:

• сохранять
• писать

Those would sound less natural here unless you meant something ongoing or habitual.

Скриншот is a masculine inanimate noun, and in Russian the accusative singular of masculine inanimate nouns is usually the same as the nominative singular.

So:

• nominative: скриншот
• accusative: скриншот

Since it is the direct object of сохранить, it is in the accusative, but the form happens to look identical.

This is very normal in Russian.

Because запрос is a masculine singular noun, so the adjective must agree with it:

• короткий = masculine singular
• запрос = masculine singular

Together they mean a short request/query.

Also, since запрос is masculine inanimate, its accusative singular looks the same as the nominative:

• nominative: короткий запрос
• accusative: короткий запрос

So even though it is the object of написать, the form does not change.

Because администратору is the dative singular form of администратор.

The dative often marks the person to whom something is written, given, sent, etc.

So:

• написать запрос администратору = write a request to the administrator

Here the administrator is the recipient of the request, so the dative is used.

Compare:

• написать другу = write to a friend
• отправить клиенту = send to the client

Not in the same way.

• администратору (dative) means to the administrator
• к администратору means to/toward the administrator in the sense of movement or direction, like go to the administrator

So:

• написать запрос администратору = correct for writing something addressed to someone
• идти к администратору = go to the administrator

These are different constructions.

Because сохранить скриншот and написать короткий запрос администратору are two coordinated infinitives in the same structure:

• попросил меня [сохранить ...] и [написать ...]

Russian normally does not put a comma before и when it simply joins two equal items in a list or two coordinated actions.

So no comma is needed here.

The whole infinitive phrase depends on попросил.

The structure is:

• Получатель письма = subject
• попросил = main verb
• меня = person asked
• сохранить скриншот и написать короткий запрос администратору = what I was asked to do

So the sentence is built like this:

[The recipient of the letter] [asked] [me] [to save a screenshot and write a short request to the administrator].

Yes. Russian word order is more flexible than English word order.

This version is neutral and natural:

• Получатель письма попросил меня сохранить скриншот и написать короткий запрос администратору.

But other orders are possible for emphasis, for example:

• Меня попросил получатель письма сохранить скриншот и написать короткий запрос администратору.
• Получатель письма попросил меня написать короткий запрос администратору и сохранить скриншот.

The basic meaning stays similar, but the focus or style changes. The original order sounds like a normal, straightforward statement.

Grammatically, запрос is fine. It usually means something like:

• request
• query
• inquiry

Depending on context, a native speaker might also use words like:

• сообщение = message
• обращение = formal appeal/request
• письмо = letter/email
• заявка = application/request ticket

But that is more about vocabulary choice than grammar. In this sentence, короткий запрос администратору is understandable as a brief written request to the administrator.