После учений пожарные показали, как пользоваться огнетушителем без паники.

Because после requires the genitive case.

The dictionary form is учения when it means drills / exercises / training exercises. After после, it changes to the genitive plural:

• nominative: учения
• genitive: учений

So:

• после учений = after the drills / after the exercises

This is a very common pattern:

• после урока = after the lesson
• после встречи = after the meeting
• после занятий = after classes

Yes, in this meaning, учения is very often used as a plural noun and refers to drills, training exercises, maneuvers as an event or set of exercises.

So после учений does not necessarily mean there were many completely separate events. It can mean after the drill session or after the exercise in a broad sense.

This is similar to how some English words are often used in a collective way.

Here пожарные is nominative plural, and it is the subject of the sentence.

The singular is:

• пожарный = firefighter

The plural is:

• пожарные = firefighters

So:

• пожарные показали = the firefighters showed

Russian often uses adjective-like profession words as nouns. Пожарный originally behaves like an adjective in form, but in modern Russian it also functions as a noun meaning firefighter.

Because the subject пожарные is plural.

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

• показал = he showed
• показала = she showed
• показало = it showed
• показали = they showed

Since пожарные = they, the verb is показали.

Here как means how and introduces an indirect question or explanatory clause:

• показали, как пользоваться огнетушителем
• they showed how to use a fire extinguisher

This is a very common Russian pattern:

• Я знаю, как это делать. = I know how to do it.
• Он объяснил, как работает машина. = He explained how the machine works.

So как пользоваться means how to use.

Both can relate to using something, but they work differently.

Пользоваться:

• usually means to use in the sense of making practical use of something
• is followed by the instrumental case
• often sounds very natural for tools, devices, services, methods

Использовать:

• is a regular transitive verb
• takes a direct object in the accusative
• often means to use / utilize

So you can say:

• пользоваться огнетушителем
• использовать огнетушитель

Both are possible, but как пользоваться огнетушителем is a very natural phrasing for explaining operation or handling.

Because пользоваться requires the instrumental case.

The dictionary form is:

• огнетушитель = fire extinguisher

Instrumental singular:

• огнетушителем

So:

• пользоваться огнетушителем = to use a fire extinguisher

This is an important verb pattern to memorize:

• пользоваться телефоном = to use a phone
• пользоваться словарём = to use a dictionary
• пользоваться компьютером = to use a computer

Literally, без паники means without panic.

In natural English, it often corresponds to:

• without panicking
• calmly
• without getting flustered

It suggests that the firefighters showed how to use the extinguisher in a calm way, not in a state of panic.

Also, без requires the genitive case, so:

• паника → паники

Most naturally, it describes how to use the extinguisher:

• They showed how to use a fire extinguisher without panicking.

So the idea is not mainly the firefighters showed it calmly, but rather they showed how one should use it calmly / without panic.

Russian can leave this kind of attachment slightly flexible, but in this sentence the most natural interpretation is that без паники belongs with пользоваться огнетушителем.

Russian often uses the infinitive in structures like this where English would say how to use.

So:

• как пользоваться огнетушителем literally is something like how to use a fire extinguisher

There is no need to add a subject such as you or one. Russian commonly leaves it general.

Compare:

• Я знаю, как готовить рис. = I know how to cook rice.
• Она показала, как открывать дверь. = She showed how to open the door.

Yes, Russian word order is flexible.

This sentence is very natural as written:

• После учений пожарные показали, как пользоваться огнетушителем без паники.

But other orders are possible, for example:

• Пожарные после учений показали, как пользоваться огнетушителем без паники.

The original order puts после учений first, which sets the scene immediately: after the drills...

That is very common in Russian. The beginning of the sentence often gives the time, setting, or context first.