ano ko ha kouen de genki ni asonde imasu.

A natural reading is:

あのこ は こうえん で げんき に あそんで います。

A few notes:

• 子 here is read こ
• 公園 is こうえん
• 遊んで is あそんで

In normal Japanese writing, you would usually see it without spaces:

あの子は公園で元気に遊んでいます。

Yes, あの子 literally means that child / that kid.

A few nuances:

• あの = that ... over there
• 子 = child / kid, but in real Japanese it can also refer to a young person, not only a small child
• Depending on context, あの子 can mean:
  • that child
  • that kid
  • that boy / that girl
  • sometimes even that person if they are being viewed as young

It does not tell you the gender by itself.

は marks the topic of the sentence.

So あの子は means something like:

• As for that child, ...
• That child ...

This makes that child the thing you are talking about.

If you used が instead, the sentence would feel more like you are identifying or emphasizing the subject:

• あの子が公園で元気に遊んでいます。
  • That child is the one playing energetically in the park.

So:

• は = topic, more general statement about that child
• が = subject focus, often more specific or contrastive depending on context

Because で marks the place where an action happens.

Here, the action is 遊ぶ = to play, so the park is the location of that action:

• 公園で遊ぶ = play at/in the park

By contrast, に is often used for:

• destination: 公園に行く = go to the park
• existence: 公園に子どもがいる = there is a child in the park
• specific time, indirect object, and some other uses

So with an action like playing, で is the correct particle.

Because 元気 is a na-adjective.

To turn a na-adjective into an adverb, you usually add に:

• 元気だ = is lively / healthy / energetic
• 元気に = energetically / cheerfully

So:

• 元気に遊ぶ = play energetically / cheerfully

Compare that with an i-adjective, which usually changes -い to -く:

• 早い → 早く
• 楽しい does not usually become 楽しく in quite the same way here, but the pattern is the standard one for i-adjectives

So 元気に is the normal adverb form.

In this sentence, 元気に most naturally means:

• energetically
• lively
• cheerfully

It comes from 元気, which often means healthy, well, or full of energy. But in context, when someone is playing, it usually means they are playing in an energetic, lively way.

So this sentence is not mainly about the child’s medical health. It is more about the way they are playing.

遊んでいます is:

• the te-form of 遊ぶ → 遊んで
• plus います

So the full structure is:

• 遊ぶ = to play
• 遊んでいる = is playing / plays regularly
• 遊んでいます = polite form of 遊んでいる

This pattern, te-form + いる / います, often expresses:

1. an action in progress
   
   • is playing
2. a habitual or repeated action
   
   • plays

In this sentence, the most likely meaning is is playing.

Because 遊ぶ is a godan verb ending in -ぶ, and verbs ending in -ぶ, -む, and -ぬ change to -んで in the te-form.

So:

• 遊ぶ → 遊んで
• 読む → 読んで
• 死ぬ → 死んで

This is just a standard te-form pattern you need to memorize.

Not always. ている / ています can mean different things depending on context.

With 遊ぶ, it can mean:

• is playing now
• is in the middle of playing
• sometimes plays regularly

For example:

• その子は毎日公園で遊んでいます。
  • That child plays in the park every day.
    Here it sounds habitual.

But without a word like every day, many learners would first understand the original sentence as:

• That child is playing energetically in the park.

Grammatically, 遊んでいます functions together as one verb phrase.

The spaces in your sentence are just there to help learners read the parts:

• あの　子　は　公園　で　元気　に　遊んで　います。

In normal Japanese, spaces are usually not used:

• あの子は公園で元気に遊んでいます。

Also, while います originally comes from the verb いる, in this pattern it works like an auxiliary attached to the te-form, giving the progressive / ongoing / habitual meaning.

Yes. It is polite because it uses います.

Compare:

• 遊んでいます = polite
• 遊んでいる = plain / casual

So:

• あの子は公園で元気に遊んでいます。
  polite
• あの子は公園で元気に遊んでいる。
  plain

The rest of the sentence is neutral, but the verb ending makes the whole sentence polite.

No. 子 often means child, but in actual Japanese it can be broader than that.

Depending on context, あの子 can refer to:

• a small child
• a school-age kid
• a teenager
• sometimes even a young adult, especially when the speaker is older

So you should not assume it always means a very young child. Context matters.

Yes, it is natural.

元気に遊ぶ is a common and natural combination, and the whole sentence sounds like a normal description of a child at a park.

It has the pattern:

• [topic] は
• [place] で
• [manner] に
• [action] ています

So it is also a nice example of standard Japanese sentence structure.