Перед ужином мама принесла поднос с тарелками и попросила меня поставить их на стол.

Because the preposition перед normally takes the instrumental case when it means before or in front of.

So:

• ужин = dinner
• перед ужином = before dinner

This is a fixed grammar pattern:

• перед работой = before work
• перед сном = before sleep
• перед уроком = before the lesson

In this sentence, перед ужином is a time expression meaning before dinner.

That is the past tense feminine singular form of the verb.

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

• он принёс = he brought
• она принесла = she brought
• оно принесло = it brought
• они принесли = they brought

Since мама is grammatically feminine, the verb must be принесла.

Принесла is the perfective past form, so it presents the action as completed: she brought it, and the result matters.

Very roughly:

• несла = was carrying / carried (focus on the process)
• приносила = used to bring / brought on some occasion(s), or brought without emphasizing completion
• принесла = brought, successfully brought and arrived with it

In this sentence, the mother completed the action and then asked the speaker to do something else, so принесла fits very naturally.

Because the preposition с here means with, and in that meaning it usually takes the instrumental case.

So:

• тарелки = plates
• с тарелками = with plates

This gives:

• поднос с тарелками = a tray with plates

Compare:

• чай с сахаром = tea with sugar
• человек с собакой = a person with a dog
• с книгами = with books

Here тарелками is the instrumental plural form of тарелки.

Because попросить in this pattern takes the person asked in the accusative case:

• попросить кого? = to ask whom?

So:

• я = I
• меня = me

That is why the sentence has:

• попросила меня поставить... = asked me to put...

A useful pattern to remember is:

• попросить кого-то сделать что-то = to ask someone to do something

For example:

• Она попросила меня помочь. = She asked me to help.
• Я попросил его подождать. = I asked him to wait.

Because поставить is perfective, and here it means to do the action once, successfully, to completion.

The contrast is:

• ставить = to put, to place, to be putting, or to do it repeatedly/habitually
• поставить = to put/place something down once, completed

After попросила меня..., Russian often uses a perfective infinitive when the speaker means do this specific action:

• попросила меня поставить их на стол = asked me to put them on the table

If you used ставить, it would sound less natural here because the request is about one completed action, not an ongoing or repeated process.

Их refers to тарелки.

That is the most natural reading because:

• тарелки is plural
• их is plural
• putting plates on the table makes sense in context

It does not naturally refer to поднос, because поднос is singular. It also does not normally mean the tray and the plates together, because Russian would usually express that differently.

So here:

• их = the plates

Because Russian uses different cases after на depending on whether you mean movement toward a place or location in a place.

Here is the key contrast:

• на стол = onto the table, to the table surface
  → accusative
• на столе = on the table
  → prepositional

In this sentence, the action is putting the plates onto the table, so Russian uses на стол.

Compare:

• Книга лежит на столе. = The book is lying on the table.
• Положи книгу на стол. = Put the book on the table.

Here it is accusative plural.

The verb поставить takes a direct object:

• поставить что? = put what?

So их is the object of поставить.

For inanimate plural nouns and pronouns, the accusative often looks like the nominative form, and with их the form stays их anyway. So the important point is grammatical function: here it is the direct object, so it is accusative.

It certainly could. Russian often uses a pronoun when the referent is already clear.

So both are possible:

• ...попросила меня поставить их на стол.
• ...попросила меня поставить тарелки на стол.

Using их sounds natural because тарелки were just mentioned. Russian avoids unnecessary repetition just like English does.

Yes. Russian word order is fairly flexible because case endings show the grammatical relationships.

The given sentence is neutral and natural:

• Перед ужином мама принесла поднос с тарелками и попросила меня поставить их на стол.

But other orders are possible, with slightly different emphasis. For example:

• Мама перед ужином принесла поднос с тарелками...
• Поднос с тарелками мама принесла перед ужином...

The original version begins with Перед ужином, which sets the time first. That is a very common way to structure a sentence.

Because перед ужином is just a normal adverbial time phrase at the beginning of the sentence, and Russian usually does not separate that with a comma.

So this is standard:

• Перед ужином мама принесла...

A comma would only appear if there were some special reason, such as parenthetical structure or a more complex insertion. In a normal sentence like this, no comma is needed.

The pattern is:

• попросить + accusative person + infinitive

So:

• попросила = asked
• меня = me
• поставить = to put

Together:

• попросила меня поставить = asked me to put

This is one of the most useful Russian sentence patterns to learn:

• Я попросил его прийти. = I asked him to come.
• Она попросила нас подождать. = She asked us to wait.
• Они попросили меня помочь. = They asked me to help.