Pater panno calido crus puellae fovet, quo ea nocte quietius dormiat.

Start with the finite verb in the main clause: fovet = he warms / he soothes / he keeps warm.

Then ask, who is doing that?
That is Pater = the father.

So the main clause is:

Pater ... fovet = The father ... warms

Everything else in that first clause helps fill in what he warms and with what.

The first clause is:

Pater panno calido crus puellae fovet

A good way to unpack it is:

• Pater = the father (subject)
• crus = leg (direct object)
• puellae = of the girl (genitive, showing possession)
• panno calido = with a warm cloth (ablative of means/instrument)
• fovet = warms

So structurally it is:

The father warms the girl's leg with a warm cloth.

Latin word order is freer than English, so you do not have to expect subject–verb–object in a fixed order.

Because it expresses the means or instrument: with a warm cloth.

This is a very common use of the ablative in Latin.

• panno = with a cloth
• calido = warm, agreeing with panno

Both are ablative singular because they go together.

So panno calido means:

with a warm cloth

not a warm cloth as a direct object.

Good question. Crus is a third-declension neuter noun. In many neuter nouns, the nominative singular and accusative singular are identical.

So:

• nominative singular: crus
• accusative singular: crus

That means crus can be the subject form or the direct-object form depending on context. Here it is the direct object because pater is clearly the subject and fovet needs something being warmed.

Puellae here is genitive singular: of the girl.

So:

• crus puellae = the girl's leg

This is a standard possessive genitive. Latin often uses the genitive where English uses 's or of.

It is not dative here. The sense is not to the girl, but the girl's.

Either order is possible in Latin. The cases tell you the relationship more clearly than the position does.

• crus = leg
• puellae = of the girl

So wherever they appear, the phrase still means the girl's leg.

That said, Latin often places related words near each other, but it does not have to follow English-style order. A learner should rely first on case endings, not on position.

Here quo introduces a purpose clause. It means something like:

• so that
• in order that
• more literally, by which / whereby

Latin very often uses quo instead of ut when the purpose clause contains a comparative, and this sentence does:

• quietius = more quietly

So:

quo ... quietius dormiat
= so that ... she may sleep more quietly

This is a standard pattern: quo + comparative + subjunctive

Because it is in a purpose clause introduced by quo.

Main idea:

• The father warms the girl's leg
• for the purpose that she sleep more quietly

Latin normally uses the subjunctive in purpose clauses.

So:

• dormit would be she sleeps (indicative, simple statement)
• dormiat = that she may sleep / might sleep (subjunctive, purpose)

Quietius is a comparative adverb meaning more quietly or more peacefully.

It comes from quietus, -a, -um = quiet, restful.

In Latin, comparative adverbs often look like the neuter singular comparative adjective:

• quietior = quieter
• quietius = more quietly

So in the clause:

quo ... quietius dormiat

the idea is:

so that ... she may sleep more quietly

Ea nocte is ablative singular, meaning:

on that night or that night

This is an ablative of time when.

• nocte = in/on the night
• ea = that, agreeing with nocte

So ea nocte does not mean she at night here. It is one phrase:

ea nocte = that night

In this sentence, the most natural reading is no. It goes with nocte:

• ea nocte = that night

Why?

Because ea and nocte match in case, number, and gender:

• feminine
• singular
• ablative

So ea is functioning as a demonstrative adjective: that.

The subject of dormiat is understood from context to be the girl.

The subject is understood, not stated again.

Latin often leaves out a subject if it is clear from context. Here, the person who is expected to sleep more quietly is obviously the girl whose leg the father is warming.

So the clause is understood as:

quo [puella] ea nocte quietius dormiat
= so that [the girl] may sleep more quietly that night

Latin does this kind of omission very often.

This is a common idiom. When a purpose clause includes a comparative idea such as:

• more easily
• more quickly
• more quietly

Latin often prefers quo rather than ut.

Examples of the pattern:

• quo facilius = so that more easily
• quo celerius = so that more quickly
• quo quietius = so that more quietly

So in your sentence, quo quietius dormiat is exactly the kind of construction students should learn to recognize as a set pattern.

You could, but it will sound less natural in English.

Very literally, quo has an underlying sense like by which or whereby. So the clause suggests:

The father warms the girl's leg with a warm cloth, whereby she may sleep more quietly that night.

That is possible English, but more natural English is:

The father warms the girl's leg with a warm cloth so that she may sleep more quietly that night.

So for reading Latin smoothly, it is usually best to take quo here as so that.

Because Latin does not need it. Once the girl has been mentioned, the sentence can simply continue with the verb dormiat, and the reader supplies the subject from context.

English usually wants an explicit subject:

• so that she may sleep

Latin often does not:

• quo ... dormiat

The omitted subject is understood as the girl.

Yes, that is a very helpful way to read it.

Part 1: Main clause

• Pater panno calido crus puellae fovet
• The father warms the girl's leg with a warm cloth

Part 2: Purpose clause

• quo ea nocte quietius dormiat
• so that she may sleep more quietly that night

So the overall logic is:

He does this in order to help her sleep better that night.

That is the key relationship between the two clauses.