V republice žije hodně lidí.

Because v meaning in normally takes the locative case when it answers where?
The basic dictionary form is republika, but its locative singular is republice.

So:

• republika = republic
• v republice = in the republic

With quantity expressions like hodně lidí (a lot of people), Czech usually uses a singular verb. The whole phrase is treated as one quantity.

So Czech says:

• Hodně lidí žije... = A lot of people live...

Compare that with:

• Lidé žijí... = People live...

Without the quantity word, the verb is plural: žijí. With hodně lidí, singular žije is normal.

After hodně, Czech uses the genitive plural.

The word for people has these relevant forms:

• lidé = nominative plural, more formal/standard
• lidi = common colloquial nominative plural
• lidí = genitive plural

Because hodně requires the genitive plural, the sentence has hodně lidí.

Here hodně means a lot of, many.

So:

• hodně lidí = a lot of people / many people

It is a very common quantity word. In other contexts, hodně can also mean a lot or very, depending on the sentence.

Yes. The whole phrase hodně lidí functions as the subject of the sentence.

Even though lidí is genitive plural, the full quantity expression acts like the subject:

• V republice žije hodně lidí.

This is one reason the verb is singular: it agrees with the quantity expression as a whole.

Czech word order is more flexible than English word order. Starting with V republice sets the scene first: In the republic...

This order is natural if the speaker wants to present the location before giving the main information.

You could also say:

• Hodně lidí žije v republice.

That is also grammatical, but the emphasis is a little different.

Czech has no articles. There is no direct equivalent of English a or the in ordinary sentences.

So v republice can mean:

• in the republic
• sometimes, depending on context, in a republic

Usually the context tells you which meaning is intended.

The verb žít can mean both to live / be alive and to live somewhere. In this sentence, because there is a place expression v republice, it clearly means to live/reside.

So:

• žije v republice = lives in the republic

Yes, but it means something slightly different.

• V republice žije hodně lidí = a lot of people live in the republic
• V republice je hodně lidí = there are a lot of people in the republic

The version with žije focuses on residence/living there. The version with je just states presence.