Hoc mense soror diem natalem celebrat, et mater convivium parvum parare incipit.

Because hoc has to agree with mense in gender, number, and case.

• mense is ablative singular
• mensis is masculine
• so the demonstrative must also be masculine ablative singular

For hic, haec, hoc, the ablative singular masculine/neuter form is hoc.

So hoc mense means this month, literally in this month.

Latin often uses the ablative of time when without any preposition.

So instead of saying in this month, Latin can simply say:

hoc mense

This is very normal with expressions of time, such as:

• hac nocte = tonight / on this night
• eo anno = in that year
• tertia hora = at the third hour

mense is ablative singular of mensis.

It is ablative because Latin commonly uses the ablative to express when something happens. Here, the sentence tells us when the sister celebrates her birthday: this month.

So:

• mensis = nominative
• mense = ablative

Because diem natalem is the direct object of celebrat.

The verb celebrat means celebrates, so it takes an object in the accusative: she celebrates what?
She celebrates diem natalem.

Also, natalem is describing diem, so it must agree with it:

• diem = accusative singular masculine
• natalem = accusative singular masculine

So both are accusative because one is the noun and the other is its agreeing modifier.

Yes. dies natalis is a standard Latin expression for birthday.

In this sentence, it appears in the accusative because it is the object of celebrat:

• nominative: dies natalis
• accusative: diem natalem

Literally it is something like day of birth, but idiomatically it just means birthday.

celebrat is present indicative active, third person singular, from celebrare.

So it means:

• she celebrates
• sometimes, depending on context, English may render it more naturally as is celebrating

The basic grammatical tense is simply present.

Because incipit often takes a complementary infinitive.

• incipit = begins
• parare = to prepare

So parare incipit means begins to prepare.

This is a very common Latin pattern:

• currere incipit = he begins to run
• dicere incipio = I begin to speak

So the infinitive after incipit works much like English to + verb.

Because mater is the subject of incipit.

The sentence has two coordinated clauses:

• soror diem natalem celebrat
• et mater convivium parvum parare incipit

In the second clause, mater is nominative, so it is the subject of incipit. The infinitive parare goes with incipit, and its understood subject is the same as the subject of incipit.

So the meaning is:

the mother begins to prepare a small feast

not that the sister does the preparing.

Yes, parvum convivium would also be possible.

Latin word order is much more flexible than English because the endings show the grammar. In this sentence:

• convivium = accusative singular neuter
• parvum = accusative singular neuter

They agree, so their relationship is clear no matter which comes first.

The order convivium parvum is perfectly natural. Latin often varies word order for rhythm, emphasis, or style rather than strict grammatical necessity.

Both are nominative singular, because both are subjects of their verbs.

• soror is the subject of celebrat
• mater is the subject of incipit

These are both third-declension feminine nouns.

So the structure is:

• soror ... celebrat
• mater ... incipit

Because Classical Latin has no articles.

Latin does not have exact equivalents of English the and a/an. Whether a noun is definite or indefinite is usually understood from context.

So:

• soror can mean the sister or a sister
• mater can mean the mother or a mother
• convivium parvum can mean a small feast or the small feast

In this sentence, English naturally supplies the articles from context.

Latin often prefers to place the verb near the end, especially in straightforward prose. So we get:

• soror diem natalem celebrat
• mater convivium parvum parare incipit

This is a very common Latin style, but it is not a rigid rule. Latin word order is flexible, and authors may change it for emphasis or elegance.

So the end-placed verbs here are normal and natural Latin.