На ферме живут кролики, козы и одна спокойная корова.

In Russian, на is commonly used with places that are thought of as areas, sites, or premises, such as:

• на ферме = on/at the farm
• на заводе = at the factory
• на рынке = at the market

Even though English says on the farm, it often really means at the farm, and Russian uses на naturally here.

Also, ферме is in the prepositional case because it comes after на when talking about location:

• ферма → basic form
• на ферме → on/at the farm

So:

• на ферме = correct for location
• в ферме would sound wrong here

Живут is the 3rd person plural form of жить (to live).

So literally, the sentence is something like:

• On the farm live rabbits, goats, and one calm cow.

Russian often uses жить for animals or people when talking about where they live. In English, we might simply say there are in some contexts, but Russian prefers живут here because these are living creatures residing on the farm.

Why plural? Because the subject is a group:

• кролики, козы и одна спокойная корова = rabbits, goats, and one calm cow

Since the whole subject is plural overall, the verb is plural:

• живут = they live

Russian often does not use a direct equivalent of English there is / there are in the present tense.

Instead of saying:

• There are rabbits on the farm

Russian very naturally says:

• На ферме живут кролики
  literally: On the farm live rabbits

So Russian often expresses existence by just stating the place and then the thing/person/animal that exists there.

This is very normal. Russian does not need a separate dummy subject like English there.

They are in the nominative plural because they are part of the subject of the sentence.

Basic singular forms:

• кролик = rabbit
• коза = goat

Plural nominative forms:

• кролики = rabbits
• козы = goats

They are the ones doing the action of живут, so nominative is the expected case.

Because корова is a feminine noun.

The numeral one changes by gender in Russian:

• один for masculine
• одна for feminine
• одно for neuter

Since корова is feminine, you must say:

• одна корова

not:

• один корова

This is a very common agreement pattern in Russian.

The adjective must agree with the noun it describes in gender, number, and case.

Here it describes корова, which is:

• feminine
• singular
• nominative

So the adjective is also:

• спокойная = feminine singular nominative

Compare:

• спокойный кролик = a calm rabbit
• спокойная корова = a calm cow
• спокойное животное = a calm animal
• спокойные козы = calm goats

So спокойная is the correct matching form for корова.

Because they are specifically modifying корова only.

The structure is:

• кролики
• козы
• и одна спокойная корова

So the sentence means:

• rabbits
• goats
• and one calm cow

It does not mean that everything in the list is one or calm. The numeral and adjective belong only to the last noun.

Because the full subject is not just корова. The subject is the whole list:

• кролики, козы и одна спокойная корова

That is a compound subject, and together it is plural in meaning. So Russian uses the plural verb:

• живут = live

If there were only the cow, it would be:

• На ферме живёт одна спокойная корова.

But with rabbits + goats + cow, it becomes plural:

• На ферме живут ...

Yes. Russian word order is more flexible than English.

This sentence:

• На ферме живут кролики, козы и одна спокойная корова.

could also be rearranged, for example:

• Кролики, козы и одна спокойная корова живут на ферме.

Both are grammatical. The difference is mainly in emphasis and style.

• На ферме живут ... puts the setting first: on the farm
• Кролики, козы ... живут на ферме puts the animals first

The original version sounds very natural when introducing what is found at the farm.

Russian has no articles like English a, an, or the.

So:

• кролики can mean rabbits or the rabbits, depending on context
• корова can mean a cow or the cow, depending on context

In this sentence, English translation naturally uses:

• rabbits, goats, and one calm cow

The numeral одна already makes корова specific as one cow, so no article is needed in Russian.

Yes, you could say:

• На ферме живут кролики, козы и спокойная корова.

That would mean:

• Rabbits, goats, and a calm cow live on the farm
  or possibly
• ... and the calm cow ..., depending on context.

Adding одна specifically emphasizes that there is one cow.

So:

• спокойная корова = a/the calm cow
• одна спокойная корова = one calm cow

This is standard list punctuation in Russian, just like in English in many cases.

In a simple list:

• кролики, козы и одна спокойная корова

you put commas between earlier items, but usually no comma before и when it means and before the final item.

So the pattern is:

• item 1, item 2 and item 3

Exactly what we see here.