suupu ga samenai uti ni, nabe ni huta wo site kudasai.

Here, うちに means something like while ... is still the case or before ... changes.

So 冷めないうちに means:

• while it hasn’t cooled yet
• before it gets cold
• before it cools down

This pattern is very common:

• 忘れないうちにメモします。
  I’ll make a note before I forget.
• 熱いうちに食べてください。
  Please eat it while it’s still hot.

In your sentence, the idea is: do the action now, before the soup loses its heat.

Because うちに often follows a negative form when you mean before something happens.

• 冷めないうちに = before it cools
• 冷めるうちに would not sound natural here

The negative form sets up the idea that the current state is still true:

• the soup has not cooled yet
• so act during that remaining time

This is a very useful pattern:

• 暗くならないうちに帰ろう。
  Let’s go home before it gets dark.
• 忘れないうちにやっておく。
  I’ll do it before I forget.

So ない + うちに is a standard and natural pattern.

Both can mean before it cools, but the nuance is a little different.

• 冷めないうちに focuses on the current state continuing for now:
  while it’s still not cool / while it’s still warm
• 冷める前に is a more direct before it cools

In many situations, they are very close in meaning.

Compare:

• スープが冷めないうちに、飲んでください。
  Drink it while it’s still warm.
• スープが冷める前に、飲んでください。
  Drink it before it cools.

The うちに version often feels a bit more like take advantage of the remaining time/state.

が marks スープ as the thing that undergoes the change 冷める.

So:

• スープが冷める = the soup cools

Here, が is natural because the sentence is talking about what may happen to the soup.

If you used は, it would sound more like setting the soup up as the topic, possibly with contrast:

• スープは冷めないうちに...

That is not impossible in some contexts, but スープが冷めないうちに is the most straightforward and natural phrasing here.

Because the sentence has two different roles:

• フタを = the thing being put on / used as the object
• 鍋に = the target place/surface it goes onto

In フタをする, フタ is marked with を because it is the thing you are placing/using as a cover.

The に after 鍋 marks where the lid goes:

• 鍋にフタをする = put a lid on the pot

This is similar to other patterns where Japanese marks both the object and the target:

• 箱に名前を書く
  write a name on the box
• 壁に絵をかける
  hang a picture on the wall

So 鍋に is not the direct object; it is the destination/target location.

Literally, it is something like do/put a lid.

In natural English, we would usually say:

• put the lid on the pot
• cover the pot

Japanese often uses Xをする in expressions where English uses a more specific verb.

So here:

• 鍋にフタをする = put a lid on the pot
• not literally do a lid in normal English

This is just a standard Japanese expression.

Actually, フタをしてください by itself sounds incomplete or unnatural here.

That is because the verb is really フタをする. To make a request, you need the て-form of する:

• する → して
• フタをしてください = please put the lid on

So the structure is:

• 鍋にフタをして = put a lid on the pot
• ください = please do so

Together:

• 鍋にフタをしてください
  Please put the lid on the pot.

Because ください attaches to the て-form of a verb to make a polite request.

Pattern:

• verb て-form + ください

Examples:

• 食べてください = please eat
• 見てください = please look
• 閉めてください = please close
• してください = please do

Since the verb here is する, its て-form is して.

So:

• フタをする → フタをして
• フタをしてください = please put on the lid

Possibly, but the nuance changes.

• 鍋にフタをしてください = put the lid on the pot
• フタをかぶせてください = place the lid over it / cover it with the lid
• フタを閉めてください = close the lid

閉める is more natural when the lid is already attached or functions like something you close, such as:

• a rice cooker lid
• a bottle cap
• a toilet lid

For a pot, 鍋にフタをする is a very common expression because it simply means put the lid on.

Japanese often leaves the subject unstated when it is obvious from context.

In a sentence ending with ください, the understood meaning is usually:

• please do this
• directed to you

So even though you is not written, the listener is clearly the person being asked to act.

This is very normal in Japanese. Subjects and pronouns are often omitted when they are easy to infer.

Yes, that is normal.

フタ means lid, and although the kanji 蓋 exists, it is often written in katakana or hiragana in everyday Japanese because the kanji is less common and a bit harder.

So all of these may appear:

• フタ
• ふた
• 蓋

In many practical or instructional sentences, フタ is very natural and easy to read.

Yes. 冷めない is the plain negative form of 冷める.

Even in polite sentences, Japanese often uses the plain form inside modifying expressions like:

• 冷めないうちに
• 食べる前に
• 雨が降ったら

The sentence becomes polite at the end with ください.

So the overall pattern is:

• clause in plain form: スープが冷めないうちに
• polite request ending: 鍋にフタをしてください

This mix is completely standard Japanese grammar.