Pallor in vultu puellae apparet, sed mater dicit eam non graviter laborare.

Because pallor is nominative singular, and it is the thing that appears: Pallor ... apparet = Paleness appears.

So the structure is:

• pallor = paleness
• apparet = appears

That makes pallor the grammatical subject.

Because in takes the ablative when it means in/on in the sense of location, and the accusative when it means into/onto with motion toward something.

Here there is no motion. The meaning is in the girl’s face, so Latin uses the ablative:

• in vultu = in the face

If it were motion, you might expect an accusative, but not here.

Puellae is genitive singular, meaning of the girl.

So:

• vultu puellae = the face of the girl / the girl’s face

The phrase in vultu puellae literally means in the face of the girl, which natural English usually turns into in the girl’s face or on the girl’s face.

A very literal order is:

• Pallor = paleness
• in vultu puellae = in the girl’s face
• apparet = appears

So literally: Paleness in the girl’s face appears.

More natural English would be:

• Paleness appears in the girl’s face
• or Paleness is visible in the girl’s face

Latin word order is more flexible than English, so the sentence does not have to follow English order to make sense.

Because after dicit, Latin is using an indirect statement construction. In this construction, the subject of the reported statement goes into the accusative, and the verb goes into the infinitive.

So:

• eam = her (accusative)
• laborare = to be suffering / to be ill (infinitive)

The sense is: the mother says her not to be suffering seriously, which in natural English becomes the mother says that she is not suffering seriously.

If you used ea, that would be nominative, and it would not fit this construction.

This is the very common Latin accusative-and-infinitive construction for indirect statement.

The parts are:

• dicit = she says
• eam = her
• non graviter laborare = not to be suffering seriously

So the literal Latin structure is:

• Mother says her not to be suffering seriously

But English usually says:

• Mother says that she is not suffering seriously

This is one of the most important Latin constructions to learn.

Because in an indirect statement after a verb like dicit, Latin normally uses an infinitive instead of a finite verb such as laborat.

Compare:

• ea non graviter laborat = she is not suffering seriously
• mater dicit eam non graviter laborare = the mother says that she is not suffering seriously

So laborare is not just randomly an infinitive; it is required by this reported-speech construction.

Not here. Although laborare can sometimes be connected with working or toiling, in many contexts it means to be in difficulty, to suffer, or especially to be unwell / to be ill.

In this sentence, because of pallor and the context of the mother commenting on the girl’s condition, laborare means something like:

• to be suffering
• to be unwell
• to be ill

So non graviter laborare means not to be seriously ill or not to be suffering badly.

Graviter means seriously, severely, or badly.

It is an adverb because it modifies the verb laborare:

• laborare = to suffer / be ill
• graviter laborare = to suffer seriously / to be seriously ill

Latin often forms adverbs from adjectives. Here gravis means heavy, serious, severe, and graviter is the adverb form.

It negates laborare within the indirect statement.

So the meaning is:

• mater dicit eam non graviter laborare
• the mother says that she is not suffering seriously

It does not mean the mother does not say... That would require the negation to apply to dicit, not to the infinitive clause.

The present infinitive in an indirect statement usually shows action happening at the same time as the main verb.

So:

• mater dicit = the mother says
• eam non graviter laborare = that she is not suffering seriously

The suffering is presented as contemporaneous with the mother’s statement.

Very roughly:

• present infinitive = happening at the same time
• perfect infinitive = happening before
• future infinitive = happening after

Yes. In English, on the girl’s face is often the most natural translation.

Latin in with the ablative can cover meanings that English may express as in or on, depending on context. With facial expressions or visible features, English often prefers on:

• Pallor in vultu puellae apparet
  = Paleness appears on the girl’s face

That is a perfectly natural way to understand it.

Because classical Latin normally reports statements after verbs like dico with the accusative-and-infinitive construction, not with that-clauses the way English does.

So English says:

• the mother says that she is not seriously ill

But Latin prefers:

• mater dicit eam non graviter laborare

This is just a normal difference between the two languages.