Pater dicit nautas, si venti mitiores fuerint, ante noctem servatos iri.

Because Latin usually reports statements with an accusative + infinitive construction rather than a that-clause.

So instead of saying literally Father says that the sailors..., Latin says something more like:

Pater dicit nautas ... servatos iri.

That is the normal way to express indirect statement after verbs like dicit, putat, scit, audit, and so on.

Because nautas is the subject of the indirect statement.

In English, the subject stays nominative in a that-clause:
Father says that the sailors will be saved.

But in Latin indirect statement, the subject is put in the accusative, so nautae becomes nautas.

A direct version would be:

Nautae ante noctem servabuntur.
= The sailors will be saved before nightfall.

When this is reported after dicit, nautae changes to nautas.

Servatos iri is the future passive infinitive of servo.

Here it means will be saved in indirect statement.

Latin uses different infinitives in indirect statement to show time relative to the main verb:

• present infinitive = action happening at the same time
• perfect infinitive = action earlier
• future infinitive = action later

Since the saving happens after dicit, Latin uses the future infinitive:

servatos iri = to be going to be saved = more natural English will be saved

Because servari is the present passive infinitive, not the future passive infinitive.

• servari = to be saved / to be being saved
• servatos iri = to be going to be saved = will be saved

So if the father is saying something about the future, servari would not be the right form.

Because it agrees with nautas.

Nautas is:

• masculine
• plural
• accusative

So servatos is also:

• masculine
• plural
• accusative

This agreement helps show that the sailors are the ones who will be saved.

Here fuerint is best understood as future perfect.

In Latin, an if-clause referring to the future often uses the future perfect, especially when English would simply use the present:

• si venerit, gaudebo
  = if he comes, I will be glad

Likewise:

si venti mitiores fuerint
= if the winds are calmer / if the winds have become calmer

So although English usually says if the winds are calmer, Latin often uses a future perfect form.

Also, the form fuerint can look like a perfect subjunctive, but here the context points to future time, so future perfect is the natural explanation.

Because venti is the subject of the finite verb fuerint inside the si-clause.

The indirect statement is:

nautas ... servatos iri

But inside it, Latin can include a normal subordinate clause:

si venti mitiores fuerint

Within that clause, venti is just the ordinary subject of fuerint, so it is nominative.

Because Latin often uses a comparative when the standard of comparison is understood.

So mitiores means milder / calmer, with the comparison left unstated:

• calmer than they are now
• calmer than before
• milder than expected

English may or may not make the comparison sound explicit, but Latin is perfectly happy to use the comparative this way.

Noctem is accusative because it follows the preposition ante, and ante takes the accusative.

So:

ante noctem = before night / before nightfall

This is just a normal prepositional phrase.

Yes. The condition belongs to the reported statement.

The sense is:

Father says that the sailors will be saved before nightfall if the winds are calmer.

So the father is not simply making one statement while the narrator adds the condition from outside. The whole conditional idea is part of what is being reported.

Because Latin word order is much freer than English word order.

The main pieces are:

• Pater dicit = Father says
• nautas = accusative subject of the indirect statement
• si venti mitiores fuerint = the condition
• ante noctem = time phrase
• servatos iri = future passive infinitive

Latin often separates words that belong together, especially in indirect statement. Once you identify the construction, the sentence becomes much easier to read.