После обеда мы играли в волейбол, и мяч несколько раз перелетал через сетку.

Because после requires the genitive case.

The noun обед changes from its dictionary form to the genitive singular:

• обед = lunch
• после обеда = after lunch

This is a very common pattern in Russian:

• после урока = after the lesson
• после работы = after work

Usually it just means after lunch in a general sense, not a specific the lunch. Russian normally does not use articles, so context decides whether something is definite or indefinite.

So после обеда is the natural way to say after lunch.

Because the verb in the past tense must agree with the subject.

Here the subject is мы = we, so the verb takes the plural past form:

• я играл / играла = I played
• он играл = he played
• она играла = she played
• мы играли = we played

So играли is the correct past plural form.

In Russian, with many games and sports, you use играть в + accusative.

So:

• играть в волейбол = to play volleyball
• играть в футбол = to play football/soccer
• играть в теннис = to play tennis

This is just the normal Russian pattern. English says play volleyball, but Russian says literally play into volleyball.

Because волейбол is an inanimate masculine noun, and in the accusative singular, many such nouns look exactly like the nominative.

So:

• nominative: волейбол
• accusative: волейбол

That is why you see в волейбол, not a different ending.

Несколько раз means several times or a few times.

• несколько = several
• раз = times

Examples:

• Я видел его несколько раз. = I saw him several times.
• Она несколько раз звонила. = She called several times.

In your sentence, it shows that the action happened more than once.

Because перелетал is imperfective, and here that makes sense because the action happened repeatedly: несколько раз.

Compare:

• перелетел = flew across/over once, completed
• перелетал = was flying across/over repeatedly / flew across several times

Since the sentence says several times, Russian normally prefers the imperfective verb.

The prefix пере- often gives the idea of crossing from one side to the other.

So:

• летать / лететь = to fly
• перелетать / перелететь = to fly across, fly over

In this sentence, the ball goes over the net, from one side to the other, so пере- is very natural.

Because через takes the accusative case.

So:

• сетка = net
• через сетку = over/across the net

The ending changes because сетка is a feminine noun:

• nominative: сетка
• accusative singular: сетку

Literally, через often means through, across, or over, depending on context.

With a volleyball net, the natural English translation is over the net. Russian still uses через because the main idea is crossing from one side to the other.

So in this sentence:

• через сетку = over the net

Russian sometimes can omit a subject if it is very clear from context, but here мяч is included to make the sentence clearer and more natural.

The sentence has two parts:

• мы играли в волейбол
• мяч несколько раз перелетал через сетку

In the second part, the focus shifts to the ball, so naming мяч is useful and natural.

Yes, Russian word order is fairly flexible. This order is neutral and natural.

The sentence starts the second clause with мяч because that is the topic of the action. Then comes несколько раз, then the verb.

Other orders are possible, for example:

• Мяч перелетал через сетку несколько раз
• Несколько раз мяч перелетал через сетку

But Мяч несколько раз перелетал через сетку sounds very normal and straightforward.

It can correspond to either one in English, depending on context.

Russian играли is past tense of an imperfective verb, so it often suggests:

• an ongoing past action: were playing
• a general past activity: played

In this sentence, После обеда мы играли в волейбол is most naturally understood as After lunch we played volleyball or we were playing volleyball after lunch. Both are possible, but English usually chooses we played volleyball here.