V této zemi žijí psi a kočky.

Because v means in, and when it expresses location in Czech, it normally requires the locative case.

So:

• země = country / land
• v zemi = in the country
• v této zemi = in this country

Both této and zemi are in the locative singular here.

Tato is the basic feminine singular form of this.

But Czech words change their endings depending on case. Since v here requires the locative, tato changes to této.

So the pattern is:

• tato země = this country
• v této zemi = in this country

This is normal adjective/demonstrative agreement: the demonstrative must match the noun in gender, number, and case.

Because země is in the locative singular after v.

Its dictionary form is:

• země = nominative singular

But in this sentence you need:

• v zemi = locative singular

So:

• země → zemi

This is a common kind of change for feminine nouns.

Žijí is the 3rd person plural present tense of žít, meaning to live.

So:

• žiji = I live
• žiješ = you live
• žije = he/she/it lives
• žijeme = we live
• žijete = you live
• žijí = they live

Because the subject is plural — psi a kočky — the verb must also be plural.

Because the subject is dogs and cats, which is plural.

• žije = he/she/it lives
• žijí = they live

Since psi a kočky means dogs and cats, Czech uses the plural verb form: žijí.

Because pes is an irregular noun.

Its nominative plural is:

• pes = dog
• psi = dogs

This is just a form you need to learn. It is one of the common irregular plurals in Czech.

So in the sentence:

• psi = dogs

Kočky is the nominative plural of kočka.

So:

• kočka = cat
• kočky = cats

Since kočky is part of the subject of the sentence, it appears in the nominative plural.

Because Czech word order is much more flexible than English word order.

English strongly prefers:

• Dogs and cats live in this country.

Czech can say that too:

• Psi a kočky žijí v této zemi.

But V této zemi žijí psi a kočky is also very natural. It puts the location first, then gives the information about what lives there.

So the sentence structure helps with focus and information flow, not just grammar.

Yes. That is grammatically correct.

Both are possible:

• V této zemi žijí psi a kočky.
• Psi a kočky žijí v této zemi.

The difference is mainly emphasis:

• V této zemi... puts focus on this country
• Psi a kočky... puts focus on dogs and cats

Yes. A means and.

So:

• psi a kočky = dogs and cats

It simply joins the two nouns together.

No. Czech does not have articles like English a/an and the.

So země can mean:

• a country
• the country
• just country

The exact meaning depends on context. In this sentence, této already makes it specific: this country.

In informal spoken Czech, yes, many speakers say žijou.

So:

• žijí = standard / written / careful speech
• žijou = common colloquial spoken form

In a textbook sentence, žijí is the expected standard form.