Learn Hardcore Russian: На первой лекции учительница по истории говорила чётким спокойным голосом. - In the first lecture the history teacher female spoke in a clear, calm voice.

So you cannot keep the adjective in nominative (первая) if the noun is in the prepositional (лекции). They have to agree: на первой лекции.

How can I tell that лекции here is singular and not plural?

The form лекции can indeed be:

singular: genitive / dative / prepositional of лекция
plural: nominative / accusative of лекции

Here context and the adjective make it clear:

первой лекции – “first lecture” can only be singular, because:
- первой is the feminine, singular, prepositional form of первая;
- you don’t usually say “at the first lectures.”

If it were plural, you’d expect something like:

на первых лекциях – at the first lectures (prepositional plural).

So на первой лекции is clearly singular: at the first lecture.

What exactly does учительница по истории mean, and what case is истории in?

Учительница по истории literally is “female teacher of history,” i.e. history teacher (female).

Preposition по here means something like “of / in the subject of”.
After по, school subjects are put in the dative case, and for история (fem. noun) the dative singular is истории.

So grammatically:

учительница – nominative, feminine, singular (subject)
по истории – по
- dative singular, indicating the subject she teaches

The pattern is common:

учитель по математике – math teacher
учительница по русскому языку – Russian language teacher

What’s the difference between учительница по истории and учитель истории?

Both refer to a history teacher, but with some nuances:

учительница по истории
- Emphasizes that the teacher is female (-ница is a feminine suffix).
- по истории is the “subject” construction (teacher of the subject history).
- Sounds more typical in everyday school context: our (female) history teacher at school.
учитель истории
- Grammatical gender is not explicit; default is masculine unless context says otherwise.
- истории here is genitive, literally teacher of history.
- Slightly more neutral/formal, often used in writing, CVs, etc.

You could say:

Моя учительница по истории очень строга. – My (female) history teacher is very strict.
Он учитель истории. – He is a history teacher.

Why is it говорила, not сказала?

Говорила and сказала both relate to “speaking,” but differ in aspect and meaning:

говорила – past tense, imperfective, from говорить
- Emphasizes the process, how she spoke over a period of time.
- Fits well with чётким спокойным голосом (clear, calm voice), describing the manner.
сказала – past tense, perfective, from сказать
- Emphasizes a single finished act of saying something.
- Typically used for a specific statement or result: she said X.

The sentence focuses on how she was speaking during the lecture, not on a particular phrase she uttered, so говорила is the natural choice.

Why is голосом in the instrumental case, and what does this construction mean?

Голосом is the instrumental singular of голос (voice).

Russian uses the instrumental case to express means or manner – by means of what / how something is done.

говорила голосом = “(she) spoke with a voice”
Adding adjectives: чётким спокойным голосом = “with a clear, calm voice” / “in a clear, calm voice.”

This is a very common pattern:

говорить тихим голосом – to speak in a quiet voice
говорить уверенным голосом – to speak in a confident voice

So говорила чётким спокойным голосом literally is “she spoke with a clear, calm voice,” describing manner of speaking.

Why are the adjectives чётким and спокойным in the instrumental, and why do they have different endings: -им vs -ым?

The adjectives must agree with голосом (instrumental, masculine, singular):

голосом (instr. masc. sg.)
чётким, спокойным (instr. masc. sg. forms)

So both are in the instrumental case because they describe голосом.

About the endings:

чёткий → чётким
спокойный → спокойным

The difference -им vs -ым is mostly spelling/phonetics, not grammar:

After К, Г, Х, Ж, Ч, Ш, Щ you usually write И, not Ы, in endings.
The stem of чёткий ends in к (чётк-), so you must write -им: чётким.
спокойный has a stem ending in н, so the normal -ым ending is used: спокойным.

Both are instrumental masculine singular forms; the difference is just the standard spelling rule.

Why are there no commas between чётким and спокойным? Could I add и (“and”)?

Here чётким спокойным голосом is just a simple series of two adjectives before a noun. In Russian:

No comma is used between two ordinary adjectives before a noun:
- чётким спокойным голосом
- маленький красный мяч – a small red ball

You can add и:

чётким и спокойным голосом

This is also correct and emphasizes them as two separate qualities slightly more explicitly, like “a clear and calm voice.” Without и, they form a more compact descriptive group.

Can the word order be changed, for example: Учительница по истории на первой лекции говорила чётким спокойным голосом?

Yes. Russian word order is relatively flexible, and several variants are possible and natural:

На первой лекции учительница по истории говорила чётким спокойным голосом.
Учительница по истории на первой лекции говорила чётким спокойным голосом.
Учительница по истории говорила на первой лекции чётким спокойным голосом.

All are grammatically correct. The differences are in focus / emphasis:

Starting with На первой лекции highlights the time/situation: “At the first lecture, the history teacher…”
Starting with Учительница по истории highlights who is acting: “The history teacher, in the first lecture, spoke…”

For a learner, the original word order is perfectly normal and neutral.

Why is it учительница … говорила, and how would it change with учитель?

In Russian, verbs in the past tense must agree with the gender of the subject.

учительница – grammatically feminine
- → говорила (past feminine)
учитель – grammatically masculine
- → говорил (past masculine)

So:

Учительница по истории говорила чётким спокойным голосом.
Учитель по истории говорил чётким спокойным голосом.

Even if in English you just say “teacher,” in Russian you must show gender in the past tense verb.

Could I say во время первой лекции instead of на первой лекции? Is there a difference?

You can say:

Во время первой лекции учительница по истории говорила чётким спокойным голосом.

This is grammatically correct, but the nuance changes slightly:

на первой лекции – at / during the first lecture in a general, natural way for events.
во время первой лекции – during the time of the first lecture, a bit more explicitly temporal and sometimes slightly more formal or “written.”

In most everyday contexts describing classroom situations, на первой лекции sounds more idiomatic and neutral.