Перед началом спектакля мы сдали пальто в гардероб и получили номерок.

This phrase contains two different case relationships:

• перед takes the instrumental case when it means before/in front of
• so начало becomes началом
• then спектакля is in the genitive case because it depends on начало: the beginning of the performance

So:

• перед
  • instrumental → перед началом
• начало чего? → спектакля

Literally, it is before the beginning of the performance.

Because начало often takes the genitive to show what is beginning.

Compare:

• начало фильма = the beginning of the film
• начало урока = the beginning of the lesson
• начало спектакля = the beginning of the performance/play

So in перед началом спектакля, the word спектакля answers beginning of what?

Сдали is the perfective past form of сдать, so it presents the action as completed.

Here the sentence describes a sequence of finished actions:

1. they checked the coat(s)
2. they got a token

So perfective verbs are natural:

• сдали
• получили

If you used сдавали, it would sound more like a process, repeated action, or background situation, not a simple completed event.

Пальто is an indeclinable noun in modern Russian. That means its form usually stays the same in all cases.

So you get:

• пальто
• без пальто
• к пальто
• с пальто

and so on.

In this sentence, пальто is the direct object, so it is in the accusative case, but the form still remains пальто.

It can be ambiguous from form alone.

Because пальто is indeclinable, the singular and plural often look the same:

• моё пальто = my coat
• мои пальто = my coats

In мы сдали пальто, context decides the meaning. In real life, it could mean:

• we checked a coat
• or we checked our coats

If the full meaning shown to the learner is we checked our coats, that is perfectly possible in Russian even though пальто itself does not change.

Russian often omits possessive words like my, your, our when the meaning is obvious from context.

So instead of saying:

• мы сдали наши пальто

Russian often just says:

• мы сдали пальто

Since it is obvious whose coat(s) are meant, the possessive is unnecessary.

Using наши is possible, but it would add emphasis or contrast.

Because в гардероб expresses motion to a place, while в гардеробе would express location in a place.

• в гардероб = into/to the cloakroom
• в гардеробе = in the cloakroom

Here the action is directional: they handed in the coat(s) to the cloakroom, so Russian uses в + accusative:

• в гардероб

Here гардероб means the cloakroom or coat check in a theater or similar venue.

In other contexts, гардероб can also mean:

• a wardrobe
• someone’s collection of clothes

But in this sentence, because people are giving in their coats before a performance, it clearly means the cloakroom / coat check.

In this context, номерок is the small numbered token/tag you receive after checking your coat.

You show it later to get your coat back.

It does not mean just any number in the abstract. Here it means a claim ticket or coat-check token.

Номерок is a common everyday word in this situation. It comes from номер with the suffix -ок, which often makes a word sound smaller or more concrete.

So номерок here means something like:

• a little numbered tag
• a small claim token

In many real-life situations in Russian, people naturally say номерок for the object you receive at a cloakroom.

Yes, Russian word order is flexible.

The sentence starts with Перед началом спектакля to set the time frame first:

• Before the performance started, ...

Then comes the main clause:

• мы сдали пальто в гардероб и получили номерок

This is a very natural order, but other orders are possible, for example:

• Мы перед началом спектакля сдали пальто в гардероб и получили номерок.

That version is also correct, but the original sounds especially natural because it places the time expression at the beginning.

Спектакль most often means a stage performance, especially a play or a theatrical production.

Depending on context, it may be translated as:

• play
• show
• performance

So in this sentence, it refers to the event in the theater before which they checked their coat(s).