Nach dem Spaziergang durch den Schlamm sind meine Sandalen ganz schmutzig.

Because German often puts a time phrase first.

Nach dem Spaziergang means after the walk and tells us when the sandals are dirty. Putting this phrase first is very natural in German.

The basic sentence without that fronted phrase would be:

Meine Sandalen sind ganz schmutzig.
= My sandals are very/completely dirty.

Then German adds the time information at the front:

Nach dem Spaziergang ... sind meine Sandalen ganz schmutzig.

So the first part is not the subject. It is just an adverbial phrase giving context.

Because nach takes the dative case when it means after.

The noun is der Spaziergang

• nominative: der Spaziergang
• dative: dem Spaziergang

So:

• nach dem Spaziergang = after the walk

This is a very common pattern:

• nach dem Essen = after the meal
• nach der Arbeit = after work
• nach dem Film = after the movie

Because durch always takes the accusative case.

The noun is der Schlamm

• nominative: der Schlamm
• accusative: den Schlamm

So:

• durch den Schlamm = through the mud

This is a fixed rule for durch. Other common accusative prepositions include für, ohne, gegen, and um.

It describes the walk.

So der Spaziergang durch den Schlamm means:

the walk through the mud

The structure is:

• Nach dem Spaziergang = after the walk
• durch den Schlamm = through the mud

Together:

• Nach dem Spaziergang durch den Schlamm = after the walk through the mud

So the speaker is not saying the sandals went through the mud separately from the walk; the phrase is part of the description of the walk.

Because German main clauses usually follow the verb-second rule.

That means the finite verb must come in the second position in the clause.

Here, the first position is taken by the whole phrase:

Nach dem Spaziergang durch den Schlamm

So the verb must come next:

sind

Then comes the subject:

meine Sandalen

So the order is:

1. Nach dem Spaziergang durch den Schlamm
2. sind
3. meine Sandalen
4. ganz schmutzig

If the subject came first, you would get:

Meine Sandalen sind nach dem Spaziergang durch den Schlamm ganz schmutzig.

That is also grammatical, just with a different emphasis.

Because the subject is meine Sandalen, which is plural.

• die Sandale = the sandal
• die Sandalen = the sandals

Plural subjects take the plural form of sein:

• ich bin
• du bist
• er/sie/es ist
• wir sind
• ihr seid
• sie sind

So:

• meine Sandalen sind = my sandals are

Because meine Sandalen is the subject of the sentence, so it is in the nominative case.

For plural nouns with mein- words:

• nominative plural: meine Sandalen
• accusative plural: meine Sandalen
• dative plural: meinen Sandalen

In this sentence, the sandals are the thing being described, so they are the subject:

Meine Sandalen sind ganz schmutzig.

If you used meinen Sandalen, that would suggest dative, which would not fit here.

Because schmutzig is a predicate adjective here, not an adjective directly in front of a noun.

Compare:

• schmutzige Sandalen = dirty sandals
  Here the adjective is before the noun, so it gets an ending.

• Die Sandalen sind schmutzig. = The sandals are dirty.
  Here the adjective comes after sein, so it does not get an ending.

In your sentence:

meine Sandalen sind ganz schmutzig

So schmutzig stays in its basic form.

Here ganz intensifies schmutzig.

So ganz schmutzig means something like:

• very dirty
• completely dirty
• really dirty

The exact nuance depends on context. In everyday speech, ganz often works as a natural intensifier.

For example:

• Die Schuhe sind ganz nass. = The shoes are really/completely wet.
• Ich bin ganz müde. = I’m really tired.

So in this sentence, ganz makes schmutzig stronger.

Yes. Sehr schmutzig is absolutely possible.

• ganz schmutzig = very/really/completely dirty
• sehr schmutzig = very dirty

Both are natural, but they can feel slightly different:

• sehr is a straightforward intensifier: very
• ganz can sometimes feel a bit more like really or entirely

In many everyday contexts, either one works.

Because in German, all nouns are capitalized.

In this sentence, the nouns are:

• Spaziergang
• Schlamm
• Sandalen

This is a basic spelling rule in German, unlike English, where common nouns are usually lowercase.

Yes. Spaziergang is historically a compound noun:

• spazieren = to stroll / go for a walk
• Gang = going / walk / passage

As a whole, der Spaziergang means walk or stroll.

German uses compound nouns very often, and the final part determines the gender. Since der Gang is masculine, der Spaziergang is also masculine.

Yes, German word order is flexible, as long as the finite verb stays in second position in a main clause.

Your sentence:

Nach dem Spaziergang durch den Schlamm sind meine Sandalen ganz schmutzig.

Another possible version:

Meine Sandalen sind nach dem Spaziergang durch den Schlamm ganz schmutzig.

Both are correct.

The difference is mainly emphasis:

• starting with Nach dem Spaziergang ... highlights the situation/time first
• starting with Meine Sandalen ... highlights the subject first

German often moves elements around for focus, but the verb-second rule still applies.