Learn Hardcore German: Im Winter brauchen wir dank des Wasserfilters keine schweren Wasserkisten mehr zu tragen. - In winter we no longer need to carry heavy boxes of water thanks to the water filter.

In normal speech and writing, Im Winter is much more natural than in dem Winter, unless you want to emphasize a very specific winter (e.g. in dem Winter 1995).

What case is Winter in, and why?

Winter is in the dative case: (in) dem Winter → im Winter.
The preposition in can take:

dative for location or time: im Haus, im Winter
accusative for movement/direction: ins Haus gehen

Here we’re talking about a time period (“in winter”), so in uses the dative.

Why is it dank des Wasserfilters and not dank dem Wasserfilter?

Traditionally, dank governs the genitive case, so des Wasserfilters is the “correct” and more formal version.
In everyday spoken German, many people also use the dative: dank dem Wasserfilter.

Both are understood; written German (especially in learning materials, newspapers, etc.) usually prefers dank + genitive: dank des Wasserfilters.

Why does Wasserfilters have an -s at the end?

The base noun is der Wasserfilter (masculine). In the genitive singular, many masculine and neuter nouns take -s or -es:

der Computer → des Computers
der Tisch → des Tisches

So der Wasserfilter becomes des Wasserfilters in the genitive. That’s why we get dank des Wasserfilters.

Why do we say keine schweren Wasserkisten and not nicht schwere Wasserkisten?

Use kein- to negate a noun phrase (“no / not any [X]”), and nicht to negate everything else (verbs, adjectives, whole clauses, specific adverbs, etc.).

Here we are negating the existence of crates as an object:

schwere Wasserkisten = heavy water crates
keine schweren Wasserkisten = no heavy water crates / not any heavy water crates

So kein(e) is correct, not nicht.

Why is it schweren and not schwere in keine schweren Wasserkisten?

Wasserkisten here is accusative plural (it’s what we don’t have to carry).
The pattern is:

article-like word keine (works like a plural article)
- adjective schwer
- noun Wasserkisten

In accusative plural with an ein/kein-word, the adjective takes -en:

keine schweren Wasserkisten
Similarly: keine alten Bücher, keine roten Äpfel.

So schweren is the correct adjective ending here.

What is the gender and case of Wasserkisten, and why is it plural?

The base word is die Kiste (feminine), plural die Kisten. When you combine it: die Wasserkiste → plural die Wasserkisten.

In the sentence, (schwere) Wasserkisten are what we (don’t) have to carry, so they’re the direct object of tragen → accusative plural.
That’s why you see keine schweren Wasserkisten (accusative plural).

How does the structure brauchen … zu tragen work? Why not müssen wir … tragen?

brauchen + zu + infinitive means “to need to do something”:

Wir brauchen Wasser zu kaufen. – We need to buy water.

In the negative, brauchen … nicht zu is very common and close to English “don’t need to / need not”:

Wir brauchen keine Wasserkisten mehr zu tragen.
Wir brauchen nicht mehr zu arbeiten.

You can also say: Im Winter müssen wir dank des Wasserfilters keine schweren Wasserkisten mehr tragen. That uses müssen (“must / have to”) with a negation of the object.
Both are correct; brauchen … nicht (mehr) zu emphasizes “there is no need to…”.

Why is zu tragen at the very end of the sentence?

German places infinitives with zu at the end of the clause.
The pattern is:

conjugated verb in second position: brauchen
everything else in the middle (subjects, objects, adverbs, prepositional phrases)
zu + infinitive at the end: zu tragen

So the structure … brauchen wir … keine schweren Wasserkisten mehr zu tragen follows the normal rule: the zu-infinitive goes to the clause-final position.

What exactly does mehr mean here, and where can it go in the sentence?

Here mehr means “any more / any longer” in a negative context:

keine … mehr = no … any more
nicht mehr = no longer / not any more

In this sentence, mehr comes after the object and before the infinitive:

keine schweren Wasserkisten mehr zu tragen

You could also say: Wir brauchen im Winter dank des Wasserfilters keine schweren Wasserkisten mehr. (with no zu tragen), and mehr would still mean “any more”. Its position is typically near the part that is being negated and before the verb or at the end of the clause.

Can I start the sentence some other way, like Wir brauchen im Winter … instead of Im Winter brauchen wir …?

Yes. German allows flexible word order as long as the finite verb stays in second position.
Both are correct:

Im Winter brauchen wir dank des Wasserfilters keine schweren Wasserkisten mehr zu tragen.
Wir brauchen im Winter dank des Wasserfilters keine schweren Wasserkisten mehr zu tragen.

In the first version, Im Winter is emphasized (when?), in the second, wir (who?) comes first. The basic meaning is the same.