Learn Hardcore German: Auf dem nassen Wanderweg sind gute Stiefel wichtig, sonst werden deine Füße schnell kalt. - On the wet hiking trail good boots are important; otherwise your feet quickly get cold.

dem = dative singular masculine (for der Wanderweg)
den would be accusative, which would imply movement onto the path (auf den Weg gehen).

Why does the adjective end with -en in dem nassen Wanderweg?

The ending -en in nassen is required by German adjective declension rules.

Wanderweg is masculine (der Wanderweg).
After auf in this sentence we use dative (see previous answer).
The definite article in dative masculine is dem.

With a definite article in dative singular (any gender), the adjective always gets -en:

dative masculine: dem nassen Wanderweg
dative feminine: der nassen Straße
dative neuter: dem nassen Feld
dative plural: den nassen Wegen

So dem nassen Wanderweg is:
article (dem) + adjective (nassen) + noun (Wanderweg) in dative masculine.

Why is it gute Stiefel and not guten Stiefel?

Gute Stiefel is nominative plural without an article, and that combination takes the adjective ending -e.

Stiefel is masculine (der Stiefel), but here it’s plural: Stiefel.
In the sentence, gute Stiefel are the subject: > … sind gute Stiefel wichtig

For nominative plural with no article, the adjective gets -e:

gute Stiefel (good boots)
alte Schuhe (old shoes)
warme Socken (warm socks)

If there were a definite article, the ending would change:

die guten Stiefel (nominative plural with article) → adjective -en
But because there is no article, we say gute Stiefel.

Why does the sentence start with Auf dem nassen Wanderweg and what does that do to word order?

German main clauses are verb‑second (V2):

Normally: Subjekt – Verb – Rest
→ Gute Stiefel sind auf dem nassen Wanderweg wichtig.

But any one element can come first, not just the subject. If you move the prepositional phrase Auf dem nassen Wanderweg to the front to emphasize the location, the verb must still stay in second position, so the subject moves after the verb:

Auf dem nassen Wanderweg (1st position)
sind (2nd position = the finite verb)
gute Stiefel wichtig (rest of the sentence)

So:

Gute Stiefel sind auf dem nassen Wanderweg wichtig.
Auf dem nassen Wanderweg sind gute Stiefel wichtig.

Both are correct; the second simply puts more emphasis on the place.

What exactly does sonst mean here, and why is there a comma before it?

In this sentence, sonst means “otherwise” or “or else”:

…, sonst werden deine Füße schnell kalt.
= “…, otherwise your feet will get cold quickly.”

Functionally, sonst is an adverb, but it acts a bit like a conjunction linking two main clauses:

Auf dem nassen Wanderweg sind gute Stiefel wichtig,
sonst werden deine Füße schnell kalt.

Because you are joining two independent main clauses, written German uses a comma before sonst.

Word order after sonst is still a normal main clause: the verb is in 2nd position (werden).

Why is it werden deine Füße schnell kalt and not just deine Füße sind schnell kalt?

German uses werden + adjective to express a change of state (become/get + adjective):

kalt werden = to get cold / to become cold
müde werden = to get tired
nass werden = to get wet

In the sentence:

…, sonst werden deine Füße schnell kalt.

the idea is: your feet will get cold (quickly), not “your feet are cold (quickly)”.

Compare:

Deine Füße sind kalt.
= Your feet are cold (state).
Deine Füße werden kalt.
= Your feet are getting cold (change).

The sentence is talking about what will happen if you don’t have good boots → change, so werden is correct.

Why is the adverb schnell placed before kalt and not after it?

In werden deine Füße schnell kalt, schnell is an adverb modifying the process of becoming cold: “get cold quickly”.

Typical position for such an adverb is:

subject – verb – adverb – predicate adjective

So:

deine Füße – werden – schnell – kalt

Putting schnell at the end, like werden deine Füße kalt schnell, sounds wrong or at least very unnatural in German. German usually keeps short adverbs like schnell, langsam, bald close to the verb or the adjective they’re modifying:

Es wird schnell dunkel. (It gets dark quickly.)
Das Wasser wird langsam kalt. (The water is slowly getting cold.)

Why is it deine Füße and not something like dein Fuß?

A few points at once:

Plural form
The noun Fuß (foot) has the plural Füße:
- singular: der Fuß
- plural: die Füße
You normally have two feet, so German uses the plural here: deine Füße.
Possessive adjective agreement
The possessive dein- has to match the case and number of the noun:
In the second clause, deine Füße is the subject → nominative plural:
- nominative plural of dein is deine:
  - deine Füße (your feet)
  - deine Hände (your hands)
You could also say, more generally:
- sonst werden die Füße schnell kalt.
  = otherwise (one’s/the) feet get cold quickly.
But if you specifically mean your feet, deine Füße is correct.

Are deine Füße in the nominative or dative here, and how can I tell?

In the clause:

sonst werden deine Füße schnell kalt.

the subject is deine Füße: your feet get cold. The subject of werden is always in the nominative.

You can test it by turning it into a simple present tense sentence:

Deine Füße sind kalt.
Wer oder was ist kalt? → deine Füße → nominative.

So deine Füße is nominative plural, not dative.

Why don’t we use an article in gute Stiefel, but we do in dem nassen Wanderweg?

Two different reasons:

Generic plural vs. specific singular
- gute Stiefel: We’re talking about good boots in general, not specific boots you already know about. German often omits the article with a generic plural:
  - Gute Stiefel sind wichtig. (Good boots are important.)
  - Warme Kleider sind teuer. (Warm clothes are expensive.)
- dem nassen Wanderweg: This is a specific trail being described by the prepositional phrase. With a singular count noun + specific reference, German normally uses an article.
Grammar of the preposition
The phrase auf dem nassen Wanderweg needs a dative object of auf. That object is realized as:
- dem nassen Wanderweg (article required here to mark the case clearly in normal style).

What is the nuance of sonst compared to oder in a sentence like this?

Compare:

…, sonst werden deine Füße schnell kalt.
…, oder deine Füße werden schnell kalt.

Both can be translated as “otherwise / or else your feet will get cold quickly”, but:

sonst (otherwise) strongly expresses a condition + consequence:
- If you don't have good boots → consequence: feet get cold.
oder (or) can sound more like just listing alternatives and is less clearly conditional; in speech it may sound a bit more casual and is often combined with a threatening tone (like “Do this, or …”).

In standard written German, sonst is the more natural choice for this kind of warning/explanation.

What’s the difference between Wanderweg and simpler words like Weg or Straße?

Weg = path/way in general.
Wanderweg = a hiking trail, a path specifically intended/used for hiking.
Straße = street/road (for cars, traffic, in towns).

So auf dem nassen Wanderweg suggests a natural trail in the countryside or mountains, not a city street.

Why is it just wichtig and not something like von Wichtigkeit or a separate verb?

In:

… sind gute Stiefel wichtig …

wichtig is a predicate adjective after the verb sein:

sein + adjective → to be + adjective
- Stiefel sind wichtig. = Boots are important.
- Das ist nötig. = That is necessary.
- Die Jacke ist warm. = The jacket is warm.

You could also say von großer Wichtigkeit, but that is more formal/literary. wichtig sein is the normal, everyday way to say “to be important”.