Mercatrix dicit Delum insulam parvam esse, sed mercatoribus utilem.

Because after dicit Latin uses an indirect statement construction.

Instead of saying:

• The merchant says that Delos is a small island

with a separate that-clause, Latin says:

• The merchant says Delos to be a small island

In this construction, the subject of the reported statement goes into the accusative, so Delos becomes Delum.

So:

• mercatrix dicit = the female merchant says
• Delum ... esse = that Delos is ...

Esse is the infinitive to be. It is the normal verb used in indirect statements after verbs like dicit.

So Delum insulam parvam esse literally means:

• Delos to be a small island

Natural English turns that into:

• that Delos is a small island

So esse is there because Latin expresses the reported statement with an accusative + infinitive construction.

They are accusative because they go with Delum inside the indirect statement.

• insulam = island
• parvam = small

Together they describe Delum:

• Delum insulam parvam esse = Delos to be a small island

Since Delum is accusative, the predicate noun and adjective that go with it are also accusative:

• Delum — accusative singular feminine
• insulam — accusative singular feminine
• parvam — accusative singular feminine

They all match.

For the same reason: it is still describing Delum in the indirect statement.

The full sense is:

• Delum insulam parvam esse, sed mercatoribus utilem [esse]

So utilem agrees with Delum:

• accusative
• singular
• feminine

because Delos is being described as useful.

Latin often leaves out a repeated word when it is easy to understand.

So:

• Delum insulam parvam esse, sed mercatoribus utilem

really means:

• Delum insulam parvam esse, sed mercatoribus utilem esse

The second esse is understood and does not need to be stated.

This is very common in Latin.

Because utilis often takes the dative in Latin.

So:

• utilem mercatoribus = useful to merchants / useful for merchants

The idea is not that merchants are being directly acted on, but that something is useful to them. That is why the dative is used.

Yes, but Latin often uses a noun like this as a predicate noun to classify or identify something.

So Delum insulam parvam esse means:

• Delos is a small island

Here insulam is not unnecessary. It tells you what kind of thing Delos is: an island.

English does the same:

• Rome is a city
• Jupiter is a planet
• Delos is an island

Mercatrix means female merchant or female trader.

It is the feminine form corresponding to mercator.

So:

• mercatrix = a woman merchant
• mercator = a male merchant, or sometimes simply merchant depending on context

This sentence interestingly contrasts:

• mercatrix — one female merchant speaking
• mercatoribus — merchants in general, in the plural

Because they refer to different people.

• mercatrix = one female merchant, the speaker
• mercatoribus = merchants in general, the people for whom Delos is useful

So the meaning is something like:

• A female merchant says that Delos is a small island, but useful for merchants.

There is no need for the two nouns to match in number.

Latin word order is much more flexible than English word order because the endings show the grammatical relationships.

This sentence puts the infinitive esse near the end, which is very normal in Latin:

• Mercatrix dicit Delum insulam parvam esse, sed mercatoribus utilem.

A more word-for-word English version would be:

• The female merchant says Delos to be a small island, but useful to merchants.

Latin uses word order more for emphasis and style than for basic grammar.

The dictionary form is utilis, utile.

It is a third-declension adjective meaning useful.

Here utilem is:

• accusative
• singular
• feminine

because it agrees with Delum, which is treated here as feminine singular.

Sed means but.

It introduces a contrast:

• parvam = small
• utilem = useful

So the idea is:

• Delos is small, but useful for merchants.

This is a nice example of Latin balancing two descriptions of the same thing.