Epistula a tabellario allata matri grata est.

Because epistula is the subject of the sentence: it is the thing being described as grata. In Latin, the subject is normally in the nominative.

• epistula = nominative singular, letter
• It is the thing that is welcome/pleasing

So the basic frame is:

• Epistula ... grata est = The letter ... is welcome/pleasing

Allata is the perfect passive participle of afferre (to bring).

Its basic idea is having been brought or simply brought.

So:

• epistula allata = the letter brought
• more fully: the letter that was brought

This participle works like an adjective and describes epistula.

Because it agrees with epistula.

Since epistula is:

• feminine
• singular
• nominative

the participle describing it must also be:

• feminine
• singular
• nominative

So we get:

• epistula allata
• not allatus or allatum

This is normal Latin agreement: adjectives and participles match the noun they describe in gender, number, and case.

Because a/ab + ablative is the normal way to express the personal agent with a passive idea.

Here, the letter is brought by the letter-carrier, so Latin says:

• a tabellario = by the letter-carrier

This is different from an inanimate means or instrument, which can often be shown by the ablative alone.

So a learner should remember:

• a/ab + ablative = by someone, especially with passive constructions

Tabellario is the ablative singular of tabellarius, meaning something like letter-carrier, messenger, or postman depending on context.

Because it follows a, it must be in the ablative:

• tabellarius = nominative
• tabellario = ablative

So:

• a tabellario = by the messenger/postman

Because gratus, grata, gratum often takes the dative of the person to whom something is pleasing, welcome, or agreeable.

So:

• matri = to the mother

The idea is not that the mother is receiving the letter as a direct object. Rather, the sentence says the letter is welcome/pleasing to the mother.

This is a very common use of the dative in Latin.

Grata est means is pleasing, is welcome, or is agreeable.

Here:

• grata is an adjective
• est is the verb is

Because epistula is feminine singular, the adjective must also be feminine singular:

• gratus = masculine
• grata = feminine
• gratum = neuter

So epistula ... grata est means the letter ... is welcome/pleasing.

In this sentence, the most natural reading is that allata is a participle modifying epistula, not the main verb of the sentence.

So the structure is:

• Epistula
• a tabellario allata
• matri grata est

That is:

• The letter, brought by the messenger, is welcome to the mother

A helpful way to see it is to expand the participle into a relative clause:

• Epistula quae a tabellario allata est matri grata est

So yes, allata est could in other contexts mean was brought, but here est is serving the main predicate grata est, while allata is adjectival.

Yes. That is often the easiest way for an English speaker to understand a participle.

• epistula a tabellario allata
  = the letter brought by the messenger
• or
  = the letter which was brought by the messenger

Latin often uses participles where English may prefer a relative clause.

Latin word order is much more flexible because the endings show how the words function.

English relies heavily on position:

• The letter brought by the messenger is welcome to the mother

Latin can move words around more freely because the forms already tell you their jobs:

• epistula = nominative subject
• a tabellario = agent
• allata = participle agreeing with epistula
• matri = dative
• grata est = main predicate

The order here is natural Latin, and it keeps related ideas close together:

• a tabellario allata goes with epistula
• matri grata est goes together as is pleasing to the mother

Latin often expresses this idea from the point of view of the thing that is pleasing, not the person who likes it.

So instead of saying:

• The mother likes the letter

Latin says something more like:

• The letter is pleasing/welcome to the mother

This is a very common Latin pattern, and English speakers often need time to get used to it. The key point is:

• the thing liked becomes the subject
• the person who likes it goes into the dative

So:

• epistula = the thing found pleasing
• matri = the person to whom it is pleasing