Pater putat canistrum a serva ad triclinium portatum esse, sed catinum adhuc in culina manere.

Because after putat (he thinks), Latin often uses an indirect statement.

In an indirect statement:

• the subject of the reported idea goes into the accusative
• the verb goes into an infinitive

So in Pater putat canistrum ... portatum esse:

• Pater putat = The father thinks
• canistrum ... portatum esse = that the basket has been carried

Here canistrum is the subject of the infinitive portatum esse, but in Latin indirect statement that subject is accusative.

Again, this is because of indirect statement after putat.

Latin does not usually say:

• Pater putat canistrum portatum est

Instead it says:

• Pater putat canistrum portatum esse

That infinitive phrase means that the basket has been carried.

More specifically:

• portatum = the perfect passive participle
• esse = to be

Together, portatum esse is the perfect passive infinitive: to have been carried.

Because portatum agrees with canistrum.

Canistrum is:

• neuter
• singular

So the participle must also be:

• neuter singular → portatum

If the noun were different, the participle would change:

• servus portatus esse
• serva portata esse
• canistrum portatum esse

The participle agrees with the accusative subject of the indirect statement.

A serva means by the slave-girl or by the female slave.

In a passive construction, the person who does the action is often expressed with:

• a/ab + ablative

So:

• a serva = by the slave-girl
• ab amico = by the friend

This is called the personal agent.

Here the sentence is passive:

• canistrum ... portatum esse = the basket to have been carried so Latin uses a serva to show who carried it.

Because ad + accusative expresses movement toward a place.

So:

• ad triclinium = to the dining room

This fits the verb portatum esse because the basket is being carried to somewhere.

Compare:

• in culina = in the kitchen → location
• ad triclinium = to the dining room → motion toward

For the same reason as canistrum: it is the subject of another indirect statement depending on putat.

The second part is:

• sed catinum adhuc in culina manere

This means:

• but that the dish is still remaining in the kitchen

So:

• catinum = accusative subject of manere
• manere = infinitive

Both parts depend on putat:

• Pater putat [canistrum ... portatum esse], sed [catinum ... manere].

In smoother English:
Father thinks that the basket has been carried by the slave-girl to the dining room, but that the dish is still in the kitchen.

Because this is still part of what the father thinks, so Latin keeps using the infinitive for indirect statement.

• manet would mean it remains
• manere means to remain

After putat, Latin wants:

• accusative subject + infinitive

So:

• catinum adhuc in culina manere = that the dish is still remaining in the kitchen

Because manere already is the verb of that clause.

In the first clause, the passive idea needs:

• portatum esse = to have been carried

But in the second clause:

• manere by itself already means to remain / to stay

So there is no need for another esse.

Latin is saying:

• the basket to have been carried
• the dish to remain

Both are normal infinitive constructions, just with different verbs.

In indirect statement, infinitive tense is usually understood relative to the main verb, not as an absolute tense in the same way as English finite verbs.

Here the main verb is putat = he thinks.

• portatum esse is a perfect infinitive, so it shows action already completed relative to putat
  → he thinks that the basket has been carried / was carried

• manere is a present infinitive, so it shows action happening at the same time as putat
  → he thinks that the dish is still remaining / is still in the kitchen

So the contrast is:

• the basket has already been taken
• the dish is still there now

Adhuc means still, up to this point, or so far.

In this sentence:

• catinum adhuc in culina manere

it shows that the dish has not yet been moved and is still in the kitchen.

It adds an idea of continuation:

• not just it is in the kitchen
• but it is still in the kitchen

Because in + ablative usually means in or on in the sense of location.

So:

• in culina = in the kitchen

This is different from in + accusative, which can show motion into a place.

Compare:

• in culina manere = to remain in the kitchen
• in culinam ire = to go into the kitchen

Here there is no movement; the dish is staying where it is, so ablative is used.

It connects two contrasting parts that both still depend on putat.

Structure:

• Pater putat
  
  • canistrum a serva ad triclinium portatum esse
  • sed catinum adhuc in culina manere

So the idea is:

• Father thinks [that the basket has been carried to the dining room], but [that the dish is still in the kitchen].

The second infinitive clause does not need another putat, because it is understood from the first part.

Yes. Serva is the feminine form.

• servus = male slave
• serva = female slave

So a serva specifically means by a female slave or by the slave-girl.