Nach der Dusche trockne ich mich mit dem Handtuch ab.

Because nach takes the dative case here.

• die Dusche = nominative / accusative
• der Dusche = dative

So:

• nach der Dusche = after the shower

This is very common:

• nach der Arbeit = after work
• nach dem Essen = after the meal / after eating

Because the speaker is drying themself.

The verb here is used reflexively:

• ich trockne mich ab = I dry myself off

The reflexive pronoun changes with the subject:

• ich trockne mich ab
• du trocknest dich ab
• er trocknet sich ab

So mich means myself here.

Ab is the separable prefix of the verb abtrocknen.

The full verb is:

• abtrocknen = to dry off / to dry oneself off / to wipe dry

In a normal main clause, German separable verbs split:

• Ich trockne mich ab.

Compare:

• infinitive: mich abtrocknen
• main clause: ich trockne mich ab

So the ab moves to the end.

Because in a main clause, separable-prefix verbs split apart.

So:

• dictionary form: abtrocknen
• conjugated main clause: ich trockne mich ab

But when the verb is in the infinitive or with a modal, it stays together:

• Ich will mich abtrocknen.
• Es ist wichtig, sich abzutrocknen.

They are related, but abtrocknen is more specific.

• trocknen = to dry / to make dry
• abtrocknen = to dry off, wipe dry

In this sentence, abtrocknen is more natural because it suggests using a towel to remove water from the body.

Examples:

• Die Sonne trocknet die Wäsche. = The sun dries the laundry.
• Ich trockne mich mit dem Handtuch ab. = I dry myself off with the towel.

Because Nach der Dusche is a time expression placed first for emphasis or natural flow.

German often puts time expressions first. But the conjugated verb still has to be in second position.

So the structure is:

• Nach der Dusche = first element
• trockne = second position
• ich = comes after the verb

You could also say:

• Ich trockne mich nach der Dusche mit dem Handtuch ab.

Both are correct.

Because mit always takes the dative case.

• das Handtuch = nominative / accusative
• dem Handtuch = dative

So:

• mit dem Handtuch = with the towel

This is another very common pattern:

• mit dem Auto = by car / with the car
• mit der Hand = with the hand
• mit den Freunden = with the friends

Mich is accusative.

In ich trockne mich ab, the reflexive pronoun is the direct object, so it is accusative.

Reflexive pronouns in the accusative:

• ich → mich
• du → dich
• er/sie/es → sich
• wir → uns
• ihr → euch
• sie/Sie → sich

A useful comparison:

• Ich trockne mich ab. = I dry myself off.
• Ich trockne mir die Haare ab. = I dry my hair.

In the second sentence, mir is dative because die Haare is the accusative object.

Usually, no—not if you want to say I dry myself off.

• Ich trockne mich ab. = correct, natural
• Ich trockne ab. = possible in some contexts, but it sounds incomplete or less standard for this meaning

If you are drying something else, you use a different object:

• Ich trockne den Hund ab. = I dry the dog off.
• Ich trockne das Geschirr ab. = I dry the dishes.

So in your sentence, mich is important.

It can naturally mean after the shower in the sense of after showering.

German often uses nouns this way. So nach der Dusche is perfectly normal.

A close alternative is:

• nach dem Duschen = after showering

Both are good, but there is a slight difference in style:

• nach der Dusche = after the shower
• nach dem Duschen = after showering

In everyday speech, both are common.

Yes.

• mit dem Handtuch = with the towel
  This suggests a specific towel, or the towel already known in context.

• mit einem Handtuch = with a towel
  This is less specific.

So both are grammatically correct. The difference is just the vs a.

Yes. trockne is present tense.

• ich trockne = I dry / I am drying

In German, the present tense is often used for both:

• what is happening now
• habits and routines

So this sentence can mean either:

• I dry myself off with the towel after the shower
  or
• After the shower, I’m drying myself off with the towel

The exact meaning depends on context.