Puer peius scribit, cum festinat; soror autem minus cito scribit et rarius errat.

Because peius is the comparative adverb of male (badly), while peior is the comparative adjective meaning worse.

Here scribit means writes, so we need a word that describes how he writes, not what kind of person he is.

• male scribit = he writes badly
• peius scribit = he writes worse / more badly

If you used peior, it would need to describe a noun, for example:

• puer peior est = the boy is worse

So in this sentence, peius is correct because it modifies the verb scribit.

Latin often forms the comparative of adverbs in two different ways:

• with a single comparative adverb, such as peius = worse
• with minus
  • an adverb, such as minus cito = less quickly

So minus cito scribit literally means she writes less quickly, which in natural English is she writes more slowly.

Latin could also express this idea with other wording in some contexts, but minus cito is a perfectly normal way to say it.

Rarius is the comparative adverb from rare or raro, meaning rarely.

So:

• raro errat = she makes mistakes rarely
• rarius errat = she makes mistakes more rarely / less often

In smoother English, that becomes she makes mistakes less often.

Like peius, rarius is an adverb because it modifies the verb errat.

Here cum means when.

• cum festinat = when he is hurrying / when he hurries

Latin uses cum with different moods depending on the meaning. In this sentence it is a straightforward time clause, so festinat is in the indicative.

A learner often notices that cum sometimes takes the subjunctive, but that usually happens in other kinds of clauses, especially when cum means something like since, although, or in more literary narrative uses.

Here the sense is simple and factual: when he hurries, he writes worse.

That is very normal for autem.

Autem means however, but, or on the other hand, and it usually comes after the first word of its clause, not at the very beginning.

So:

• soror autem = the sister, however / but the sister

This may feel unusual to an English speaker, because English normally puts but/however first. Latin often prefers this second-position placement with words like autem and enim.

Latin often can leave out repeated words, but it does not always do so.

Here the repetition makes the contrast very clear:

• Puer peius scribit
• soror autem minus cito scribit et rarius errat

Repeating scribit helps set up the comparison between the boy and the sister:

• he writes worse
• she writes less quickly

Then the sentence adds another point about her: and she makes mistakes less often

So the repetition is not strange; it is clear and natural.

Because Latin usually does not need subject pronouns when the verb ending already shows the person and number.

For example:

• scribit = he/she writes
• festinat = he/she hurries
• errat = he/she makes mistakes / goes wrong

The nouns puer and soror supply the subjects, so Latin does not need extra words for he and she.

English usually requires the pronoun, but Latin often does not.

Both puer and soror are nominative singular, because they are the subjects of their verbs.

• puer ... scribit = the boy ... writes
• soror ... scribit = the sister ... writes

A native English speaker may not think much about case, but in Latin the subject is usually in the nominative. The verb endings also support this:

• scribit = he/she writes
• errat = he/she makes mistakes

So the nominative forms puer and soror fit their role as subjects.

Yes. Literally, errat means he/she errs, goes wrong, or makes a mistake.

In English, she errs less often is correct but sounds formal or old-fashioned. So learners are often shown the more natural English:

• rarius errat = she makes mistakes less often

That is a translation choice for natural English style. The Latin verb itself is a single word: errat.

The sentence contrasts the boy and his sister in two different ways.

About the boy:

• Puer peius scribit, cum festinat
  = he writes worse when he is in a hurry

About the sister:

• soror autem minus cito scribit
  = but his sister writes less quickly / more slowly
• et rarius errat
  = and she makes mistakes less often

So the basic idea is:

• the boy is faster, but his writing gets worse when he rushes
• the sister is slower, but more accurate

This kind of contrast is signaled especially by autem and by the comparative adverbs peius, minus cito, and rarius.

Because the sentence is describing actions, not nouns.

Adverbs modify verbs:

• scribit = writes
• festinat = hurries
• errat = makes mistakes

So the sentence tells us how the actions happen:

• peius = worse
• minus cito = less quickly
• rarius = less often

If Latin were describing the boy or the sister directly, then adjectives would be used instead. But here the focus is on writing badly, writing less quickly, and making mistakes less often, so adverbs are the right choice.