Learn Hardcore German: Zu Hause haben wir einen neuen Wasserfilter für das Leitungswasser gekauft. - At home we have bought a new water filter for the tap water.

German main clauses usually keep the verb in second position (the V2 rule). Only one element can stand before the conjugated verb, and here that element is Zu Hause (an adverbial of place).

So we get:

Zu Hause haben wir einen neuen Wasserfilter … gekauft.
– place first, then verb in 2nd position.

You can definitely also say:

Wir haben zu Hause einen neuen Wasserfilter … gekauft.

Both are correct.
The difference is emphasis:

Zu Hause haben wir … gekauft. → slightly emphasizes where you bought it.
Wir haben zu Hause … gekauft. → more neutral; subject wir in first position.

What exactly is Zu Hause grammatically, and why is Hause capitalized?

Zu Hause is a fixed expression meaning at home.

zu is a preposition.
Hause is an old dative form of das Haus (house), so it is a noun and must be capitalized.

So:

zu Hause = at home (adverbial phrase of place)

You may also see Zuhause written as one word when it is used as a noun:

Ich mag mein Zuhause. – I like my home.

In your sentence, Zu Hause functions as an adverbial (where?), so it is normally written as zu Hause (two words). At the beginning of a sentence, Zu is capitalized only because it’s the first word.

Why is the verb haben in second position and gekauft at the end?

The sentence is in the present perfect tense (Perfekt):

auxiliary verb: haben (conjugated: haben wir)
past participle: gekauft

In German:

The conjugated auxiliary (here haben) must be in second position in a main clause.
The past participle (here gekauft) goes to the end of the clause.

Structure:

Zu Hause – element in first position
haben – conjugated verb in second position
wir einen neuen Wasserfilter für das Leitungswasser – middle part
gekauft – past participle at the end

So the word order follows standard Perfekt rules: V2 + participle at the end.

Why do we use haben and not a simple past form like kauften?

In everyday spoken German, people almost always use the Perfekt (haben/sein + participle) for past events:

Wir haben einen neuen Wasserfilter gekauft.

The simple past (Präteritum) like wir kauften is:

common in written language (books, reports),
common in speech mostly with a few verbs (e.g. war, hatte, konnte, musste).

So haben … gekauft is the natural, conversational way to say we bought.
Wir kauften einen neuen Wasserfilter is correct but sounds formal or written.

Why is it einen neuen Wasserfilter and not ein neuer Wasserfilter?

Wasserfilter is masculine: der Wasserfilter.

In the sentence, it is the direct object (what did we buy?), so it is in the accusative case.

Masculine with ein in the accusative takes -en on both the article and the adjective:

Nominative: ein neuer Wasserfilter (subject)
- Ein neuer Wasserfilter ist teuer. – A new water filter is expensive.
Accusative: einen neuen Wasserfilter (object)
- Wir kaufen einen neuen Wasserfilter. – We are buying a new water filter.

So:
einen (accusative masc.) + neuen (adjective ending -en) + Wasserfilter (noun).

Why does neu become neuen in einen neuen Wasserfilter?

The adjective neu has to agree with:

the gender: masculine
the case: accusative
the article type: ein (indefinite article)

For ein + masculine + accusative, the adjective takes -en:

einen neuen Wasserfilter

Pattern (masc. singular with ein):

Nominative: ein neuer Filter
Accusative: einen neuen Filter
Dative: einem neuen Filter
Genitive: eines neuen Filters

So neu → neuen is simply the correct adjective ending in this context.

Why is für das Leitungswasser in the accusative?

The preposition für always takes the accusative case.

So no matter what comes after für, you must use accusative:

für den Mann (der Mann → den Mann)
für die Frau (die Frau → die Frau)
für das Kind (das Kind → das Kind)
für das Leitungswasser (das Leitungswasser → das Leitungswasser)

Here:

das Leitungswasser is neuter; in accusative it stays das.
The case is accusative because of für, not because of a subject/object role.

What does Leitungswasser literally mean, and how is it formed?

Leitungswasser is a compound noun:

die Leitung – pipe / line (here: water pipe)
das Wasser – water

Literally: pipe water or water from the pipes, which corresponds to tap water in English.

Because it’s a compound, only the last part determines gender:

das Wasser is neuter → das Leitungswasser is also neuter.

Is the word order einen neuen Wasserfilter für das Leitungswasser fixed, or can I move für das Leitungswasser?

German word order in the middle field (between the conjugated verb and the final verb) is flexible.

You could say:

Zu Hause haben wir einen neuen Wasserfilter für das Leitungswasser gekauft. (very natural)
Zu Hause haben wir für das Leitungswasser einen neuen Wasserfilter gekauft.

Both are grammatically correct.
Typical tendencies:

Short, closely connected elements (einen neuen Wasserfilter) often come before longer or more “optional” details (für das Leitungswasser).
Putting für das Leitungswasser earlier slightly emphasizes what the filter is for.

Why is Wasserfilter capitalized?

In German, all nouns are capitalized.

Wasserfilter is a noun (a thing: a filter), so it must begin with a capital letter:

der Wasserfilter – the water filter

This rule applies even in the middle of a sentence, unlike in English.

Can I leave out Zu Hause without making the sentence ungrammatical?

Yes, you can omit Zu Hause if the context already makes the location clear:

Wir haben einen neuen Wasserfilter für das Leitungswasser gekauft.

That is a perfectly correct sentence.
Adding Zu Hause simply adds the information where you bought it and allows you to emphasize that place by putting it first:

Zu Hause haben wir … gekauft. → At home, we bought …