Vilicus mane ad agrum properat, dum vilica apud villam manet et operas servorum disponit.

Yes. They are closely related.

• vilicus is a masculine noun meaning farm steward, overseer, or bailiff
• vilica is the feminine form, meaning female steward, housekeeper, or overseer of the household side of the estate

So the sentence contrasts the male and female managers of the estate:

• vilicus handles one side of the work
• vilica handles another

They are both in the nominative singular here because each is the subject of its own verb.

mane is an adverb, and it means in the morning or early in the morning.

It is not the subject or object of a verb. It simply tells us when the action happens:

• Vilicus mane ad agrum properat = The steward hurries to the field in the morning

A learner might expect something more like in the morning as a phrase, but Latin often uses a single adverb where English uses a prepositional phrase.

Because ad takes the accusative case.

• dictionary form: ager = field
• accusative singular: agrum

So:

• ad agrum = to the field

This is a very common Latin pattern:

• ad
  • accusative = movement toward something

That fits the verb properat well, since he is hurrying to the field.

This is a very useful contrast.

• ad agrum = to the field
  This shows motion toward a place.
• apud villam = at / near the farmhouse or estate house
  This shows location near or at a place, not motion toward it.

So the sentence sets up a contrast:

• the vilicus goes to the field
• the vilica remains at the villa

Even though both ad and apud are followed by the accusative, they do not mean the same thing:

• ad often suggests direction
• apud suggests position at, among, in the presence of, near

Because some Latin prepositions always take a particular case, regardless of whether English would use motion or rest.

apud always takes the accusative.

So:

• dictionary form: villa
• after apud: apud villam

This can feel strange to an English speaker, because English often links case-like ideas to motion versus location. Latin sometimes does that, but not always. With apud, you simply learn:

• apud
  • accusative = at, near, among, in the presence of

Here dum means while.

It connects the two parts of the sentence:

• Vilicus mane ad agrum properat
• dum vilica apud villam manet et operas servorum disponit

So the sense is:

• The steward hurries to the field, while the stewardess remains at the villa and organizes the slaves’ work.

A learner may know that dum can also mean until in some contexts. That is true. But here, with present-tense verbs describing simultaneous actions, while is the natural meaning.

They are in the present indicative active, third person singular:

• properat = he hurries
• manet = she remains / stays
• disponit = she arranges / organizes

Latin often uses the present tense to describe a scene in a simple, straightforward way. English may translate these as simple present or sometimes as more natural continuous English depending on context:

• hurries / is hurrying
• remains / is staying
• arranges / is arranging

In a basic teaching sentence like this, the present tense simply shows what each person is doing.

Because it shows possession or association: of the slaves.

• dictionary form: servus = slave
• genitive plural: servorum

So:

• operas servorum = the work/services/tasks of the slaves

In smoother English, this may become:

• the slaves’ tasks
• the work of the slaves
• the slaves’ labor

The genitive is very often the Latin case for of.

operas is the accusative plural of opera.

In this context, opera means things like:

• work
• effort
• service
• labor
• sometimes more concretely, tasks

Because it is the direct object of disponit, it is in the accusative:

• operas disponit = she arranges/organizes the tasks/work

So operas servorum disponit means that the vilica is organizing the labor or duties of the slaves.

disponit comes from dispono, disponere, which means to arrange, set in order, distribute, organize.

So operas servorum disponit suggests more than just seeing the work; it implies that she is:

• assigning it
• ordering it
• planning it
• organizing how it will be done

A good natural translation is often:

• she organizes the slaves’ work or
• she assigns the slaves’ tasks

Latin has a much freer word order than English because the endings show grammatical function.

For example:

• vilicus is the subject because of its form and role
• agrum is the object of ad
• servorum is genitive
• operas is the direct object of disponit

Because of this, Latin does not rely as heavily on word order to show meaning.

This sentence is actually fairly straightforward, but it still does not match rigid English order exactly. Latin word order often helps with:

• emphasis
• balance
• style
• contrast

Here the structure neatly contrasts the two figures:

• Vilicus ... properat
• vilica ... manet et disponit

et simply means and. It joins two actions done by the same subject, vilica:

• manet = she remains
• disponit = she arranges

So:

• vilica apud villam manet et operas servorum disponit
• the vilica remains at the villa and organizes the slaves’ work

Latin often leaves a shared subject unstated after the first mention, just as English does:

• she remains and organizes

Yes. The sentence is built around a clear contrast:

• properat = hurries
• manet = remains / stays

So one person is moving quickly away to the field, while the other stays at the villa and manages work there. This gives the sentence a neat division of responsibilities:

• outdoor/agricultural action for the vilicus
• household or estate management for the vilica

This kind of balanced contrast is very common in elementary Latin sentences because it helps learners see the grammar clearly.