Наши места в партере были рядом со сценой.

Наши is the plural nominative form of наш (our).

It agrees with места, which is plural here:

• singular: наше место
• plural: наши места

Because the subject of the sentence is места (seats), the possessive adjective has to match it in number, gender (where relevant), and case.

Here места is nominative plural of место.

You can tell it is plural because:

• it is paired with наши rather than наше
• the verb is были, which is plural

Compare:

• Наше место было рядом со сценой. = one seat/place
• Наши места были рядом со сценой. = more than one seat/place

So in this sentence, места definitely means seats/places in the plural.

Были is the past tense plural form of быть (to be).

In Russian past tense, the verb agrees with the subject in number and, in the singular, also in gender:

• был = masculine singular
• была = feminine singular
• было = neuter singular
• были = plural

Since места is plural, the sentence uses были.

In the present tense, Russian often leaves out to be:

• Наши места рядом со сценой.

But in the past tense, you normally must use the past form of быть:

• Наши места были рядом со сценой.

So you cannot normally omit были in this sentence.

Because в can take different cases depending on the meaning.

Here it means in or in the stalls, expressing location, so Russian uses the prepositional case:

• в партере

If there were motion into somewhere, Russian would often use the accusative:

• идти в театр
• сесть в партер can sometimes be used in the sense of going/taking seats there

But in your sentence, the seats are already located there, so в партере is correct.

Партере is the prepositional singular of партер.

Партер is a masculine noun, and many masculine nouns take -е in the prepositional singular after в or на when talking about location:

• в театре
• в зале
• в партере

So в партере means in the stalls / in the parterre section.

Сценой is the instrumental singular form of сцена.

The expression рядом с / со means next to / beside, and it requires the instrumental case:

• рядом с домом
• рядом с театром
• рядом со сценой

So:

• сцена = basic form
• сценой = instrumental singular, used after рядом с/со

Russian sometimes uses со instead of с to make pronunciation easier.

This often happens before awkward consonant clusters. Here с сценой would put several difficult consonants together, so Russian prefers:

• со сценой

This is similar to:

• со мной
• со всеми
• со школы in some contexts

So the reason is mainly pronunciation and sound flow, not a difference in meaning.

It is often best understood as an adverb-like word that commonly works together with с / со + instrumental.

So in:

• рядом со сценой

the full expression means next to the stage.

You will often see:

• рядом с домом
• рядом с нами
• рядом со мной

Learners often just memorize it as a pattern:

• рядом с/со + instrumental

That is the safest approach.

Yes. Russian word order is fairly flexible, and the basic meaning would stay the same.

For example:

• Наши места в партере были рядом со сценой.
• В партере наши места были рядом со сценой.
• Наши места были рядом со сценой, в партере.

The original order sounds natural because it first identifies which seats are being talked about, then says where they were.

Russian word order often changes for:

• emphasis
• topic vs. new information
• style

So the sentence is not locked into only one possible order.

Yes, место literally means place, but it is also very commonly used to mean a seat or an assigned place, especially in contexts like:

• theatres
• cinemas
• trains
• airplanes

So наши места can naturally mean:

• our seats
• our places

This is very normal Russian usage.

The dictionary form is сцена.

It is a feminine noun, which you can see from the ending -а in its basic form. Its instrumental singular is:

• сценой

This pattern is very common for feminine nouns ending in -а:

• комната → комнатой
• машина → машиной
• сцена → сценой

So со сценой follows a regular declension pattern.

Yes, it could work, but it is not exactly the same.

• рядом со сценой = right next to / beside the stage
• около сцены = near the stage

Also, the grammar changes:

• рядом с/со + instrumental → со сценой
• около + genitive → сцены

So both can be correct depending on the intended nuance, but рядом со сценой sounds more specifically beside the stage.