Learn Hardcore Russian: Пожалуйста, покажите пропуск охране у входа. - Please show your pass to security at the entrance.

Покажите is the imperative (command/request) form of the verb показать (to show). Specifically, it’s the polite/plural imperative, used when addressing:

one person formally (you, polite вы), or
multiple people (you all).

The informal singular would be покажи.

What’s the difference between показать and показывать here?

This is about aspect:

показать (perfective) = show once / complete the action (what you want at a checkpoint).
показывать (imperfective) = be showing / show repeatedly or habitually.

At an entrance, you normally use perfective: Покажите пропуск… (Show your pass [now].)

Why isn’t вы (you) written in the sentence?

Russian often omits subject pronouns because the verb ending already shows who is addressed. Покажите already signals you (polite/plural), so вы is unnecessary unless you want extra emphasis/contrast.

What case is пропуск, and why?

Пропуск is in the accusative case because it’s the direct object of покажите (show what?).
For an inanimate masculine noun like пропуск, accusative = nominative, so it stays пропуск.

Why is it охране and not охрану or охранник?

Охране is dative case (to/for someone), because the idea is show something to someone:

показать (что?) пропуск (кому?) охране.

Охрана means security (as a service/group). If you mean one guard, you could say:

Покажите пропуск охраннику (dative singular of охранник).

What does у входа mean грамmatically, and what case is входа?

у means by/near/at (location). It requires the genitive case, so:

вход → у входа (genitive singular).

So у входа = at/near the entrance.

Is the word order fixed? Could I move parts of the sentence around?

The sentence is natural as-is. Russian word order is flexible, but changes can add emphasis:

Пожалуйста, покажите пропуск охране у входа. (neutral)
У входа покажите пропуск охране. (focus on location: “At the entrance…”)
Охране у входа покажите пропуск. (focus on “the security at the entrance”)

What exactly does пропуск mean in everyday Russian?

Пропуск is typically an entry pass / access badge / permit—something that allows you to enter a building/area. It’s common in offices, factories, events, or guarded buildings. (A passport is паспорт, and a ticket is билет.)

How formal/polite is this sentence? Is Пожалуйста required?

It’s polite and fairly standard for signs or staff instructions.
Пожалуйста softens the command into a request. You can omit it in stricter/official wording, but it will sound more blunt:

Покажите пропуск охране у входа. (still normal, just less softened)

Could I also say предъявите пропуск instead of покажите пропуск?

Yes. Предъявите пропуск is a very common official/checkpoint phrasing meaning present your pass.
Compared to покажите, предъявите sounds more formal/official (like security or administrative language).

How do I pronounce it (stress)?

Main stresses:

пожа́луйста
покажи́те
про́пуск
охра́не
у вхо́да

Approximate rhythm: Пожа́луйста, покажи́те про́пуск охра́не у вхо́да.