Pater consilium medicae approbat et dicit puellam domi manere oportere.

Pater is clearly the subject because it is nominative singular.

Consilium is the direct object of approbat, so here it is accusative singular. Since consilium is a neuter noun, its nominative and accusative singular look the same, so you have to use syntax and meaning to tell which it is.

So:

• pater = the father, doing the action
• consilium = the plan/advice, receiving the action of approbat

Here medicae is best understood as genitive singular, meaning of the doctor.

So consilium medicae means:

• the doctor's plan
• the doctor's advice

The ending -ae could also be dative singular in other contexts, but here the genitive makes much better sense.

Yes. Medica is the feminine form, so it means female doctor or woman physician.

Its genitive singular is medicae, which is why the sentence means the female doctor's advice/plan.

If the doctor were male, Latin would usually use:

• medicus = male doctor
• medici = of the male doctor

In this sentence, consilium means something like:

• plan
• advice
• recommendation
• decision

A learner might connect it with English council, but that is not the right meaning here. In Latin, consilium very often refers to an idea, plan, or piece of advice rather than a governing group.

So consilium medicae approbat means that the father approves the doctor's recommendation or plan.

Because Latin usually does not use a that-clause after verbs of saying, thinking, knowing, and so on.

Instead, it often uses indirect statement, which has this pattern:

• verb of saying/thinking
• accusative
• infinitive

So after dicit, Latin gives:

• puellam ... oportere

This is the Latin way of expressing what English would say with that:

• He says that the girl ought to stay at home.

There are two helpful ways to think about this.

First, after dicit, Latin uses indirect statement, where the subject of the reported statement appears in the accusative.

Second, oportet is an impersonal verb, and the person who is obliged is often put in the accusative anyway.

So:

• puellam domi manere oportet = the girl ought to stay at home
• after dicit, oportet becomes oportere
• puellam stays accusative

That is why you see puellam, not puella.

Because the sentence contains a layered construction.

The main verb is dicit = he says.

What he says is expressed by an indirect statement:

• puellam ... oportere

But oportere itself needs another infinitive to show what action is necessary:

• manere = to stay/remain

So the structure is:

• dicit
• puellam ... oportere
• manere

In natural English:

• He says that the girl ought to stay at home.

In very literal English:

• He says the girl to ought to stay at home.

That literal version sounds bad in English, but it helps show how Latin is built.

Because the clause after dicit is in indirect statement, and in indirect statement Latin uses an infinitive instead of a finite verb.

The normal main-clause form is:

• oportet = it is proper / it is necessary / one ought

But in indirect statement it changes to:

• oportere = to be proper / to be necessary / to ought

So:

• direct statement: Puellam domi manere oportet
• reported after dicit: dicit puellam domi manere oportere

Domi means at home.

It is an example of the locative, an old case that survives in a few Latin words, especially:

• names of towns and small islands
• a few special nouns such as domus

So:

• domi = at home
• domum = homeward, to home
• domo = from home

This is why domi does not look like a regular ablative or accusative form.

Because domi is the normal idiomatic way to say at home.

There is a slight difference in feel:

• domi = at home
• in domo = in the house, inside the building

So if the idea is simply that the girl should stay home, domi manere is more natural.

Latin word order is flexible because the endings show the grammatical roles. This sentence is arranged in a fairly clear way:

• Pater = subject
• consilium medicae = object phrase
• approbat = first main verb
• et dicit = second main verb
• puellam domi manere oportere = what he says

So the sentence breaks down as:

• Pater consilium medicae approbat
• et dicit puellam domi manere oportere

Latin could move some of these words around for emphasis, but the meanings would still be shown mainly by the endings and the construction.