Pater vult filium ad piscinam non currere, ne in frigidarium cito cadat.

Because filium is the subject of the infinitive currere, not the subject of vult.

• pater = the one who wants
• filium = the one who is supposed not to run

After verbs like vult, Latin often uses an infinitive for the action wanted. If the person doing that action is different from the main subject, Latin commonly puts that person in the accusative.

So:

• Pater vult currere = The father wants to run
• Pater vult filium currere = The father wants the son to run

In your sentence, filium is therefore accusative.

Because it depends on vult.

Latin often uses volo + infinitive to say want to do or want someone to do something. Here currere is the action that the father wants or does not want.

So the structure is:

• vult ... currere = wants ... to run

Since the father wants his son not to run, Latin uses:

• vult filium non currere

Here non negates currere, not vult.

So the idea is:

• vult filium non currere = he wants his son not to run

If Latin wanted to say the father does not want his son to run, it could also negate vult:

• Pater non vult filium currere

That is slightly different in emphasis:

• vult filium non currere = he wants the son not to run
• non vult filium currere = he does not want the son to run

In practice, these can be very close in meaning, but the negation is attached to a different part of the sentence.

Because ne introduces a negative purpose clause.

Here the idea is not simply he does not fall, but rather so that he may not fall or lest he fall. Latin normally uses:

• ut
  • subjunctive for positive purpose
• ne
  • subjunctive for negative purpose

So:

• ut cadat would not make sense here
• ne cadat = so that he may not fall / lest he fall

non is usually used to negate a word, phrase, or ordinary statement, not to introduce a negative purpose clause.

Because it is in a clause introduced by ne, and purpose clauses in Latin take the subjunctive.

So:

• cadit = he falls or he is falling
  This would be a plain statement.
• cadat = may fall / might fall

In this sentence, the father wants his son not to run so that he may not fall, so Latin uses the subjunctive cadat.

The understood subject is filium.

Latin often leaves out a pronoun when the subject is obvious from context. No new subject is introduced in the ne clause, so the natural understanding is that the same person is meant:

• the son is the one who might fall

So ne in frigidarium cito cadat means that the son might quickly fall into the frigidarium.

Because Latin is making two slightly different ideas of motion:

• ad
  • accusative = to / toward
• in
  • accusative = into

So:

• ad piscinam = to the pool, possibly just toward it
• in frigidarium = into the cold room, emphasizing entry into it

Latin often uses ad for motion toward a person or place, and in with the accusative for movement into an enclosed space or interior.

Both are accusative singular.

They are accusative because:

• ad takes the accusative
• in takes the accusative when it means into and shows motion

So:

• ad piscinam → accusative after ad
• in frigidarium → accusative after in of motion into

If in meant in or inside without motion, it would usually take the ablative instead.

Because frigidarium is a neuter second-declension noun.

In neuter second-declension nouns:

• nominative singular = -um
• accusative singular = -um

So the nominative and accusative look the same.

That means:

• nominative: frigidarium
• accusative: frigidarium

Even though the form has not changed, the case is still accusative here because in with motion takes the accusative.

It most naturally goes with cadat.

Because it is placed right before cadat, the most likely sense is:

• ne in frigidarium cito cadat = lest he quickly fall into the frigidarium

Latin word order is flexible, but placement often helps show what an adverb most closely modifies. Here cito is most naturally understood with the falling, not the running.

Yes, to a large extent.

Latin word order is much freer than English word order because the endings show the grammatical relationships. So a sentence like this could be rearranged for emphasis without completely changing the meaning.

For example, Latin can move words around to highlight:

• the person involved
• the negation
• the destination
• the feared result

But the exact order used here is quite natural:

• Pater first: the topic
• vult filium ... currere: the main wish
• ne ... cadat: the purpose or feared result

So the order is not random, but it is more flexible than English.