У дома есть маленькая клумба, за которой мама ухаживает каждое утро.

Here у дома means by the house / near the house.

Literally, у + the genitive case often means at, by, near. So:

• у дома есть маленькая клумба = there is a small flowerbed by the house

In English, this may also be translated more naturally as the house has a small flowerbed, but the Russian structure is really presenting the flowerbed as something located near the house.

Because the preposition у requires the genitive case.

So:

• дом = nominative
• дома = genitive singular
• в доме would mean in the house, because в takes a different case here

So у дома is correct because у + genitive = by/near the house.

That is a very common question.

Russian usually does omit the present-tense copula is/are in sentences like:

• Это мой дом = This is my house

But есть in this sentence is not just a plain is. It has an existential meaning: there is / there exists / there are.

So:

• У дома есть маленькая клумба = There is a small flowerbed by the house

If you remove есть, the sentence may still be understandable in some contexts, but есть clearly emphasizes the existence or presence of the flowerbed.

Because клумба is a feminine singular noun in the nominative case, and the adjective must agree with it.

So:

• клумба = feminine singular nominative
• маленькая = feminine singular nominative to match it

This is the thing that exists in the sentence, so it appears in the nominative:

• есть маленькая клумба = there is a small flowerbed

За которой means which or for which / that ... for it, depending on how literally you think about it.

The noun being referred to is клумба, so the relative pronoun must match клумба in:

• gender: feminine
• number: singular

That is why we get которой, the feminine singular form of который.

But its case is not chosen by клумба. Its case is chosen by its role inside the relative clause:

• мама ухаживает за клумбой
• therefore: за которой мама ухаживает

So которой agrees with клумба in gender and number, but takes the case required by за in this clause.

No. In this sentence, за does not mean behind.

That is because ухаживать за is a fixed expression meaning:

• to take care of
• to look after
• to tend

So:

• мама ухаживает за клумбой = Mom takes care of the flowerbed

Even though за often can mean behind, here it is simply part of the verb phrase ухаживать за.

Because the verb phrase ухаживать за requires the instrumental case.

Compare:

• клумба = nominative
• клумбу = accusative
• клумбой = instrumental

You would say:

• мама ухаживает за клумбой

So when that noun is replaced by the relative pronoun, it must also be instrumental:

• за которой

If it were которую, that would be accusative, and it would not fit the grammar of ухаживать за.

Because за которой мама ухаживает каждое утро is a relative clause that describes клумба.

Russian puts a comma before this kind of clause, just like English often separates relative clauses in writing, though the punctuation rules are not always exactly the same.

So the structure is:

• main clause: У дома есть маленькая клумба
• relative clause: за которой мама ухаживает каждое утро

The comma marks the start of that descriptive clause.

Каждое утро is in the accusative case, used as an adverbial expression of time.

Russian often uses the accusative to express every X, all day, every week, and similar repeated time expressions.

So:

• утро is neuter
• каждое agrees with it
• каждое утро = every morning

Compare:

• каждый день = every day
• каждую неделю = every week
• каждое утро = every morning

Yes. Russian word order is more flexible than English word order.

This sentence uses a natural, neutral order:

• У дома есть маленькая клумба, за которой мама ухаживает каждое утро.

But other orders are possible, especially in the relative clause, for emphasis or style. For example, you might also see:

• ..., за которой мама каждое утро ухаживает
• ..., за которой каждое утро мама ухаживает

These all mean roughly the same thing, but the focus shifts slightly depending on what the speaker wants to emphasize.

Yes. Russian could also say something like:

• У дома есть маленькая клумба. Мама ухаживает за ней каждое утро.

That means:

• There is a small flowerbed by the house. Mom takes care of it every morning.

The version with за которой is just a more connected, elegant way to combine the two ideas into one sentence.