Стоит дочке взять пластилин, как она начинает лепить смешных животных.

This is a common Russian pattern meaning something like:

• as soon as...
• the moment...
• all it takes is for ... to ... and then ...

So:

Стоит дочке взять пластилин, как она начинает лепить смешных животных.

means roughly:

• As soon as the daughter gets hold of some modeling clay, she starts making funny animals.
• The moment she takes the modeling clay, she starts sculpting funny animals.

The idea is that the first action immediately triggers the second one.

Because this construction usually puts the person who performs the infinitive in the dative:

• стоит кому-то сделать..., как...

So here:

• дочке = to the daughter / for the daughter
• взять = to take

Literally, the structure is something like:

• It only takes for the daughter to take the modeling clay, and...

This is not how English normally says it, but it helps explain why the dative is used.

Compare:

• Стоит мне открыть книгу, как я засыпаю.
  As soon as I open a book, I fall asleep.
• Стоит ему позвонить, как все начинают волноваться.
  As soon as he calls, everyone starts worrying.

No. Here стоит is from стоить, not стоять.

So this is not:

• stands

It is the verb стоить, which usually means:

• to cost
• to be worth

But in this special construction, стоит + infinitive, как..., it means something like:

• it only takes...
• one has only to...

So стоит дочке взять пластилин is idiomatic, not literal.

Because взять is perfective, and here Russian wants a single completed action that triggers the next event.

• взять = to take / pick up once, successfully
• брать = to take / be taking / take habitually

In this sentence, the meaning is:

• the moment she takes the modeling clay,
• she starts sculpting.

That is why взять fits well.

If you used брать, it would sound less natural here because the sentence is about a specific trigger event, not an ongoing or repeated process in itself.

No, here как does not mean how.

In this pattern, как introduces the second event after the trigger in the first clause. It works almost like:

• and then
• when
• as soon as

So:

• Стоит дочке взять пластилин, как она начинает лепить...

means:

• As soon as the daughter takes the modeling clay, she starts...

This use of как is tied to the whole construction стоит..., как...

начинает лепить means starts sculpting / begins to make.

This emphasizes the immediate beginning of the action after she takes the clay.

So the sentence is focusing on the chain:

1. she takes the modeling clay
2. she immediately starts making funny animals

If you said лепит смешных животных, that would mean simply she sculpts/makes funny animals. That is possible in some contexts, but начинает лепить makes the reaction feel more immediate and vivid.

Also note the aspect:

• начинает is imperfective, because starting is presented as a process/state
• лепить is also imperfective, because it refers to the activity of sculpting

Because смешных животных is the accusative plural of an animate noun phrase.

The verb лепить takes a direct object, so животных is in the accusative.

In Russian, for animate plural nouns, the accusative looks like the genitive:

• nominative plural: смешные животные
• accusative plural: смешных животных

So even though it may look like genitive, here it is actually accusative plural, because these are the things she is making.

Compare:

• Я вижу смешных животных.
  I see funny animals.
• Она лепит смешных животных.
  She is sculpting funny animals.

But with inanimate nouns:

• nominative plural: смешные предметы
• accusative plural: смешные предметы

So the form changes here because животные are animate.

дочке is the dative singular of дочка.

A few useful points:

• дочка is a common, warm, everyday word for daughter
• дочь is the more basic dictionary form
• дочери is the dative of дочь
• дочке is the dative of дочка

So:

• дочке sounds natural, affectionate, and conversational
• дочери would sound more formal or neutral-bookish in many contexts

Because the sentence sounds like everyday speech about a child, дочке is very natural.

Пластилин is in the accusative singular, because it is the direct object of взять.

But it looks unchanged because it is:

• masculine
• inanimate

For many masculine inanimate nouns, the accusative singular is the same as the nominative singular:

• nominative: пластилин
• accusative: пластилин

So the form does not visibly change.

Yes, at least to some extent, because this is a fairly fixed construction.

The sentence begins with:

• Стоит дочке взять пластилин...

That immediately signals the idiom as soon as / the moment.

Russian word order is flexible, but fixed expressions often have their own preferred shape. In this case, starting with Стоит... is the most natural way to build this pattern.

If you changed the order too much, the sentence could sound awkward or even be understood differently.

Yes. A very common simpler equivalent would be:

• Как только дочка берёт пластилин, она начинает лепить смешных животных.
• Как только дочка возьмёт пластилин, она начинает лепить смешных животных.

Or more naturally:

• Как только дочка берёт пластилин, она сразу начинает лепить смешных животных.

The original version with Стоит..., как... is a bit more expressive and idiomatic. It gives the sense that the reaction is automatic:

• the moment clay appears in her hands, she is off making funny animals.

So the original is not especially difficult once you recognize it as a set pattern.