Learn Hardcore Russian: Ведущая спокойно говорит с публикой. - The host speaks calmly with the audience.

Ведущая is grammatically a feminine form of an adjective/participle (from the verb вести – “to lead”), but in this sentence it is used as a noun meaning “(female) host / presenter / anchor”.
Russian often turns adjectives and participles into nouns to name people (for example работающий → “worker”).
The ending -ая marks it as feminine singular, nominative case, so ведущая literally means “the (female) leading one” and by usage “the female host / presenter”.

How would the sentence change if the host were male instead of female?

You would use the masculine form ведущий:

Ведущий спокойно говорит с публикой.

Only the word ведущая → ведущий changes; the rest of the sentence stays the same.
Ведущий is the masculine singular nominative form and means “(male) host/presenter”.

Why is there no word like “the” or “a” before ведущая?

Russian has no articles (no direct equivalents of “a/an” or “the”).
So ведущая on its own can mean “a host” or “the host” depending on context.
The listener works out whether it’s specific or general from the situation, not from a separate word.

What does спокойно mean here, and what form is it?

Спокойно is an adverb meaning “calmly”.
It comes from the adjective спокойный (“calm”) and answers the question как? (“how?”):
говорит как? → спокойно (“speaks how?” → “speaks calmly”).
Adverbs like спокойно do not change for gender, number, or case.

Why is it говорит and not the infinitive говорить?

Говорить is the infinitive (“to speak / to talk”).
In the sentence you need a conjugated verb to match the subject ведущая (3rd person, singular):

я говорю
ты говоришь
он/она говорит ← used here

So ведущая говорит = “the host speaks / is speaking”.

Why do we use с with публикой? Could we say “to the audience” instead?

The verb pattern here is говорить с кем? → “to talk with someone”, and с (“with”) always takes the instrumental case.
So говорить с публикой means “to talk with the audience” (implying some interaction).

If you want “talk to the audience”, Russian often uses:

говорить публике (дательный падеж / dative) – “talk to the audience”
or обращаться к публике – “address the audience”

Why is публикой in that form? Which case is it?

The base noun is публика (“public, audience”), feminine.
In the sentence we have публикой – this is instrumental case, singular.
The preposition с (“with”) requires the instrumental case when it means “together with / in company with”:

с другом – with a (male) friend
с подругой – with a (female) friend
с публикой – with the audience

What is the difference between говорить с публикой and говорить публике / говорить перед публикой?

говорить с публикой – “to talk with the audience”: suggests a two-way interaction, conversation, Q&A, etc.
говорить публике – “to talk to the audience”: emphasis on one-way speaking to them (dative case).
говорить перед публикой – “to speak in front of the audience”: focuses on the physical situation of speaking before an audience, like giving a speech or performance, not necessarily interactive.

All are correct, but they highlight different aspects of the situation.

How is the present tense говорит translated – “speaks” or “is speaking”?

Russian has only one present tense for actions in the present.
Говорит can correspond to both English forms:

“The host speaks with the audience.”
“The host is speaking with the audience.”

Context decides which English form sounds more natural; in Russian the verb form stays говорит.

Can we move спокойно to a different place in the sentence?

Yes. The most neutral word order is the original:

Ведущая спокойно говорит с публикой.

You can also say:

Ведущая говорит с публикой спокойно. – stylistically a bit heavier, often adding emphasis to how she speaks.
Спокойно ведущая говорит с публикой. – more poetic/literary or with strong focus on спокойно.

The meaning (“the host speaks calmly with the audience”) stays essentially the same; word order mainly affects emphasis and style.

How would the sentence look in the plural, with several hosts?

For multiple (mixed-gender or male) hosts:

Ведущие спокойно говорят с публикой.

Changes:

ведущая → ведущие (plural subject)
говорит → говорят (3rd person plural verb form)

If you wanted specifically several women, you would still normally use ведущие in the plural.

How do you pronounce the whole sentence and where is the stress in each word?

Stressed syllables are marked in capitals in the approximated transcription:

Ведущая – ведУщая – ve-DOO-sha-ya
спокойно – споко́йно – spa-KÓY-na
говорит – говорИт – ga-va-REET
с – s (like English “s”)
публикой – пУбликой – POO-bli-koy

Whole sentence (slowly):
ведУщая спако́йно говорИт с пУбликой.