Ведущая спокойно говорит с публикой.

to speak

Breakdown of Ведущая спокойно говорит с публикой.

• с: with
• спокойно: calmly
• публика: the audience

Questions & Answers about Ведущая спокойно говорит с публикой.

What exactly is ведущая here – a noun, an adjective, or a participle, and why is it feminine?

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.