Nach dem Duschen nehme ich den Föhn und kämme meine Haare mit einem Kamm.

Because Duschen is being used as a noun here, not just as a verb.

In German, an infinitive can be turned into a noun, similar to English showering or the act of showering. When that happens, it is capitalized:

• duschen = to shower
• das Duschen = showering / the act of showering

So nach dem Duschen literally means after the showering, but in natural English we would just say after showering or after taking a shower.

Because nach is a preposition, and here it is followed by a noun phrase. Since Duschen has been turned into the noun das Duschen, it needs an article.

Also, nach takes the dative case, so:

• das Duschen → dem Duschen in the dative

That is why you get:

• nach dem Duschen

So the structure is:

• nach
  • dative noun phrase

Because German main clauses follow the verb-second rule.

The finite verb must come in the second position of the clause. If you begin with a time phrase like Nach dem Duschen, that whole phrase takes the first position, so the verb comes next:

• Nach dem Duschen | nehme | ich den Föhn.

This is very common in German:

• Heute gehe ich...
• Am Morgen trinke ich...
• Nach dem Duschen nehme ich...

So even though ich is the subject, it does not have to come first.

They are in different cases because they do different jobs in the sentence.

• den Föhn is the direct object of nehme
  → direct objects usually take the accusative
• einem Kamm comes after mit
  → mit always takes the dative

So:

• der Föhn → den Föhn (masculine accusative)
• ein Kamm → einem Kamm (masculine dative)

That gives:

• ich nehme den Föhn
• mit einem Kamm

German usually uses the plural die Haare when talking about the hair on someone's head.

So:

• meine Haare = my hair

The singular Haar often means a single hair:

• Ich habe ein Haar auf dem Pullover. = I have a hair on my sweater.

You can sometimes see das Haar in the singular to mean hair in a general or stylistic sense, but in everyday speech die Haare is very common when talking about someone's hair.

Yes. In fact, ich kämme mir die Haare is often more idiomatic in German.

German very often uses:

• a dative pronoun for the person
• plus a definite article for the body part

For example:

• Ich wasche mir die Hände.
• Ich putze mir die Zähne.
• Ich kämme mir die Haare.

So both of these are correct:

• Ich kämme meine Haare.
• Ich kämme mir die Haare.

The version with mir die Haare can sound a bit more natural because it follows a very common German pattern for actions done to your own body.

You are right: kämmen already implies using a comb, so mit einem Kamm is somewhat redundant.

The sentence is still grammatical, but native speakers would often leave that part out unless they want to emphasize the tool:

• Ich kämme meine Haare. = normal, natural
• Ich kämme meine Haare mit einem Kamm. = emphasizes that I am using a comb

You might say it if you want to contrast it with something else:

• ...mit einem Kamm, nicht mit einer Bürste.

So it is correct, but not always necessary.

Both can mean roughly the same thing, but the grammar is different.

nach dem Duschen

• nach
  • noun phrase
• concise and compact
• literally after the showering

nachdem ich geduscht habe

• nachdem is a conjunction
• followed by a full clause
• means after I have showered

Compare:

• Nach dem Duschen nehme ich den Föhn.
• Nachdem ich geduscht habe, nehme ich den Föhn.

Both are correct. The first is shorter and very common. The second is more clause-based.

Also notice the spelling:

• nach dem = two words
• nachdem = one word

They are not interchangeable word-for-word.

The noun is der Föhn, so it is masculine.

In this sentence, der Föhn means hair dryer or blow dryer.

That is why the accusative form is:

• den Föhn

So:

• der Föhn = the hair dryer
• ich nehme den Föhn = I take the hair dryer

A learner may also notice that Föhn can refer to a warm wind in geography/weather contexts, but in everyday household language it commonly means hair dryer.