Секретарь секции проверил список.

Because it is in the genitive singular: секции = of the section.

In Russian, one noun often modifies another by putting the second noun in the genitive case:

• секретарь секции = secretary of the section
• literally, secretary [of] section

So секретарь is the main noun, and секции tells you which secretary.

Here секретарь is in the nominative singular, because it is the subject of the sentence — the person who performed the action.

So in this sentence:

• секретарь = subject
• проверил = verb
• список = direct object

The -л is the normal marker of the past tense in Russian.

The past tense is also marked for gender and number:

• проверил = masculine singular
• проверила = feminine singular
• проверило = neuter singular
• проверили = plural

So проверил tells you that the subject is understood as masculine singular.

Yes, in this sentence it does.

The noun секретарь can refer to either a male or a female secretary, depending on context. But the verb form проверил is masculine, so here it means the secretary is male.

If the secretary were female, you would say:

• Секретарь секции проверила список.

It is in the accusative. The important point is that for many inanimate masculine nouns in Russian, the accusative singular looks exactly like the nominative singular.

So:

• nominative: список
• accusative: список

That is why the form does not change.

Compare this with an animate masculine noun, where nominative and accusative are often different.

Проверил is perfective.

That means the action is viewed as completed: he checked the list, finished checking it.

The imperfective partner is проверял, which would suggest an ongoing, repeated, or process-focused action:

• проверил = checked, finished checking
• проверял = was checking / checked regularly / checked as a process

In this sentence, the perfective form presents the action as a completed event.

The dictionary form is проверить.

That is the infinitive of the perfective verb. So you can think of the forms like this:

• infinitive: проверить
• past masculine: проверил

A learner should usually memorize verbs with their aspect, so it is useful to know:

• проверить = perfective
• проверять = imperfective

Russian normally has no articles.

So a sentence like this can mean:

• The secretary of the section checked the list
• or, depending on context, A secretary of the section checked a list

English requires articles, but Russian usually leaves that information to context.

Yes, Russian word order is fairly flexible, because case endings carry much of the grammatical information.

The sentence as written —

• Секретарь секции проверил список.

— is a normal, neutral order.

Other orders are possible, but they change the emphasis or style. For example:

• Список проверил секретарь секции.
  This puts more focus on the list or contrasts who checked it.

Even though word order can vary, the version you were given is the most straightforward neutral statement.

Because this is a very common Russian noun-plus-genitive pattern.

When one noun defines another, Russian often uses:

• main noun + genitive noun

So:

• секретарь секции = secretary of the section
• директор школы = director of the school
• номер комнаты = number of the room

Even though секции could have other case functions in other sentences, here its role after секретарь clearly makes it a genitive relationship.

The word stress is:

• секрета́рь
• се́кции
• прове́рил
• спи́сок

A rough pronunciation guide:

• секрета́рь ≈ syekrye-TARʹ
• се́кции ≈ SYEK-tsii
• прове́рил ≈ pra-VYER-il
• спи́сок ≈ SPEE-sak

A full rough pronunciation of the sentence would be something like:

• syekrye-TARʹ SYEK-tsii pra-VYER-il SPEE-sak

The soft sign in секретарь does not have its own sound; it makes the preceding р soft.

The soft sign does not represent a separate sound by itself. Its job is to soften the preceding consonant.

So in секретарь, the final р is pronounced softly.

This is also just part of the normal spelling of this noun in the nominative singular:

• секретарь

Many masculine nouns ending in a soft consonant or soft-sign pattern work this way.

They are:

• секретарь — noun, masculine
• секция — noun, feminine
• проверить — verb, perfective
• список — noun, masculine

And their forms in the sentence are:

• секретарь — nominative singular
• секции — genitive singular
• проверил — past tense, masculine singular
• список — accusative singular form, identical here to nominative because it is inanimate masculine