Haec facultas magna est: puella nunc potest philosopham audire et quaestiones ponere.

Because haec has to agree with facultas in gender, number, and case.

• facultas is feminine
• singular
• nominative

So the correct form is haec, the feminine nominative singular of hic, haec, hoc.

Even though facultas ends in -as, it is not masculine; it is a third-declension feminine noun.

Facultas is nominative singular. In this sentence it is the subject of est:

• Haec facultas magna est = This opportunity/ability is great

A good clue is that it is the thing being described by magna and linked to est.

Magna is an adjective modifying facultas, so it must agree with it:

• facultas = feminine, singular, nominative
• therefore magna = feminine, singular, nominative

So haec facultas magna est means literally This opportunity is great.

Facultas can mean things like:

• opportunity
• chance
• ability
• possibility
• sometimes privilege

The exact English word depends on context. In this sentence, it likely means something like opportunity or possibility, since the next part explains what the girl is able to do.

The colon is not doing anything specifically “Latin-grammatical”; it is punctuation chosen by the writer.

Here it introduces an explanation of why this is a great opportunity:

• Haec facultas magna est:
• puella nunc potest philosopham audire et quaestiones ponere.

So the second part explains the first part.

Because puella is the subject of potest.

• puella = the girl
• potest = is able / can

So puella nunc potest... means the girl can now...

It is also understood as the subject of both infinitives:

• audire
• ponere

After possum (I am able / I can), Latin normally uses an infinitive.

So:

• potest audire = she can hear/listen to
• potest ponere = she can put/ask

This is very similar to English can + verb.

The subject of the infinitives is the same as the subject of potest, namely puella.

Because philosopham is accusative singular, the direct object of audire.

• philosopha = female philosopher (nominative singular)
• philosopham = female philosopher (accusative singular)

Since the girl is hearing/listening to the philosopher, Latin puts philosopha into the accusative:

• puella ... philosopham audire potest

It can cover both ideas, depending on context.

• hear = perceive with the ears
• listen to = pay attention to what someone is saying

In a sentence about hearing a philosopher and asking questions, English often prefers listen to, but Latin uses audire quite naturally here.

Because it is the direct object of ponere.

• quaestio = question (singular)
• quaestiones = questions (accusative plural here)

The girl can ask questions, so questions is the thing being “put” or “posed.”

Its basic meaning is indeed to put or to place. But Latin often uses ponere in expressions where English uses a more idiomatic verb.

So:

• quaestiones ponere = literally to put/place questions
• natural English = to ask questions

This is a very normal kind of idiom in Latin.

Latin word order is more flexible than English word order. Nunc means now, and it is placed before potest for emphasis or natural flow:

• puella nunc potest... = the girl can now...

You could move nunc to other positions and still keep the basic meaning, though the emphasis might shift slightly.

Yes, but not in the same rigid way as in English.

Latin uses endings to show grammatical relationships, so word order is often used more for:

• emphasis
• style
• clarity
• rhythm

For example:

• puella nunc potest philosopham audire et quaestiones ponere

clearly presents the subject first, then now, then the verb, then the two infinitive phrases. It is a neat and natural order.

Here et joins the two actions the girl can do:

• philosopham audire
• quaestiones ponere

So the structure is:

• puella nunc potest
  • philosopham audire
  • et quaestiones ponere

In other words, she can now listen to the philosopher and ask questions.

Because audire and ponere are infinitives, and Latin infinitives already include the idea that English often expresses with to + verb.

So:

• audire = to hear / to listen to
• ponere = to put / to ask (in this phrase)

Latin does not add a separate word corresponding to English to here.