Gemella mea matrem ad forum sequitur, dum ego cum avia domi maneo.

Because sequitur comes from sequor, which is a deponent verb.

Deponent verbs:

• use passive-looking forms
• but have active meanings

So:

• sequor = I follow
• sequitur = he/she follows

It does not mean is followed.

Matrem is the direct object of sequitur.

The person doing the action is gemella mea.
The person being followed is matrem.

So:

• mater = mother (nominative)
• matrem = mother as a direct object (accusative)

Even though sequor is deponent, it still takes a normal accusative object.

Gemella is nominative singular feminine, so it works as the subject of sequitur.

It agrees with mea, which is also:

• nominative
• singular
• feminine

So gemella mea is the subject phrase.

The verb sequitur is 3rd person singular, which matches a singular subject.

Because ad means to or toward, and it takes the accusative case.

So:

• ad forum = to the forum

This shows motion toward a place.

A learner may compare:

• ad forum = to the forum
• in foro = in the forum

So ad is used because the mother is being followed to the forum, not simply located there.

It actually is accusative, but forum is a neuter second-declension noun, and in neuter nouns:

• nominative singular = accusative singular

So:

• nominative: forum
• accusative: forum

The form looks the same, but the function is different.
After ad, it must be understood as accusative.

Here dum means while.

It connects the two actions and shows that they happen at the same time:

• Gemella mea ... sequitur
• dum ego ... maneo

So the sense is while my twin sister follows mother to the forum, I stay at home with grandmother.

In other contexts, dum can also mean until or as long as, but here while is the natural meaning.

Yes. The ending of maneo already tells you the subject is I.

So Latin could simply say:

dum cum avia domi maneo

But ego is added for emphasis or contrast.

That gives the sense of:

• while I, on the other hand, stay at home with grandmother

Latin often includes subject pronouns when the writer wants to make a contrast clearer.

Because the preposition cum meaning with takes the ablative case.

So:

• avia here is ablative singular
• cum avia = with grandmother

This is a very common rule:

• cum amico = with a friend
• cum puella = with a girl
• cum avia = with grandmother

Domi is a special form meaning at home.

It is called the locative form, and a few words in Latin keep this old location form.
Domus is one of the most common examples.

So:

• domi = at home

This is more idiomatic than in domo if you simply mean at home.

By contrast:

• in domo can mean more literally in the house
• domi means at home as a general location

Latin word order is much more flexible than English word order.

So both of these are possible:

• mea gemella
• gemella mea

Both can mean my twin or my twin sister, depending on context.

Latin often places possessive adjectives either before or after the noun. The choice can be stylistic, rhythmic, or slightly emphatic, but in a basic sentence the meaning usually stays the same.

Gemella is the feminine form of a word meaning twin.

In context, it is often translated as:

• my twin sister
• or more literally, my female twin

It can function almost like a noun by itself, even though historically it is related to an adjective.

So gemella mea is a natural Latin way to say my twin sister / my twin.

Because they are simply two different kinds of verbs.

• maneo, manere is a regular active verb: I remain / I stay
• sequor, sequi is a deponent verb: passive form, active meaning

So:

• maneo = normal active form
• sequitur = deponent form with active meaning

A learner has to memorize which verbs are deponent and which are not.

Not as important as in English.

Latin uses:

• case endings
• verb endings
• prepositions

to show how the words fit together.

So the core meaning would stay the same even if the order changed, for example:

Dum ego cum avia domi maneo, gemella mea matrem ad forum sequitur.

That said, word order still affects emphasis and flow.
The given version puts gemella mea first, then shifts to ego in the second clause, which helps highlight the contrast between the two people.