В следующей главе героиня находит первую улику в старом письме.

Because after в meaning in, Russian usually uses the prepositional case for location.

So:

• глава = nominative, dictionary form
• в главе = in the chapter
• в следующей главе = in the next chapter

The adjective has to match the noun, so both words change:

• следующая глава = next chapter
• в следующей главе = in the next chapter

Here, both следующей and главе are feminine singular prepositional forms.

Because следующий is an adjective, and adjectives in Russian must agree with the noun they describe in:

• gender
• number
• case

The noun глава is:

• feminine
• singular
• here in the prepositional case

So the adjective must also be feminine singular prepositional:

• nominative: следующая глава
• prepositional: в следующей главе

That -ей ending is a normal feminine singular ending for this case.

Because героиня is the subject of the sentence, and subjects are normally in the nominative case.

In this sentence, героиня is the one doing the action of finding, so nominative is correct:

• героиня = the heroine
• героиню would be accusative, used if she were the direct object instead

So:

• героиня находит = the heroine finds

Находит is present tense, from the verb находить.

Russian often uses the present tense in summaries, plot descriptions, and narration, much like English can do in phrases such as In the next chapter, the heroine finds...

This is sometimes called the narrative present.

So even if the story is fictional or refers to events in the plot, present tense is very natural here.

The verb находить is imperfective, while найти is perfective.

In plot summaries and general narration, Russian often uses the imperfective present:

• героиня находит улику = the heroine finds a clue

A true present-tense form of the perfective verb usually refers to the future:

• героиня найдёт улику = the heroine will find a clue

So находит fits well because the sentence is presenting what happens in that chapter, not emphasizing a one-time completed future event.

Because улика is the direct object of the verb находит, so it goes into the accusative case.

Since улика is a feminine noun, both the noun and adjective change:

• nominative: первая улика
• accusative: первую улику

This is a very common pattern with feminine nouns ending in -а:

• книга → книгу
• новая книга → новую книгу
• улика → улику
• первая улика → первую улику

Because after в meaning in, Russian uses the prepositional case for location, and письмо changes to письме.

So:

• письмо = letter
• в письме = in the letter
• в старом письме = in the old letter

The adjective also agrees:

• старое письмо = old letter
• в старом письме = in the old letter

Here старом and письме are neuter singular prepositional forms.

Usually it means in the letter, that is, in the contents of the letter.

Russian uses в письме very naturally for something written in a letter. Depending on context, it could also be understood more literally, but the normal reading is:

• the clue is found in an old letter
• meaning the clue is contained in what the letter says

So it works much like English in a letter.

Yes. Russian word order is relatively flexible because case endings show grammatical roles.

The sentence as written:

В следующей главе героиня находит первую улику в старом письме.

is natural and neutral. It starts with the setting in the next chapter, then gives the subject, then the verb, then the object, then extra detail.

Other orders are possible, for example:

• Героиня находит первую улику в старом письме в следующей главе.
• Первую улику героиня находит в старом письме в следующей главе.

But changing the order changes the focus or emphasis. The original sentence is a very natural way to present a plot event.

Russian does not have articles.

English uses articles to mark things like:

• the heroine
• a clue
• the next chapter

Russian usually expresses that information through:

• context
• word order
• adjectives
• the general situation

So:

• героиня can mean the heroine
• улику can mean a clue or the clue, depending on context
• первую улику often strongly suggests the first clue

You learn to infer this from the sentence and the surrounding context, rather than from articles.

A helpful stress guide is:

В слЕдующей главЕ героИня нахОдит пЕрвую улИку в стАром письмЕ.

A few important stress points:

• слЕдующей
• главЕ
• героИня
• нахОдит
• пЕрвую
• улИку
• стАром
• письмЕ

Stress is important in Russian because it is not always predictable and can affect vowel pronunciation.