Mein Vater schraubt das Regal an die Wand fest.

Because the verb here is the separable verb festschrauben.

• Full infinitive: etwas festschrauben = to screw something tight / to fasten something with screws

• In a main clause, the conjugated part goes to position 2, and the separable prefix goes to the end:

• Ich schraube das Regal fest.
• Mein Vater schraubt das Regal an die Wand fest.

So:

• schraubt = the conjugated verb part
• fest = the separated prefix

This is very common in German with separable verbs.

No. In this sentence, an is not part of the separable verb.

The separable verb is festschrauben.

The words an die Wand form a prepositional phrase:

• an = to / onto / against
• die Wand = the wall

So the structure is:

• Mein Vater = subject
• schraubt ... fest = verb festschrauben
• das Regal = direct object
• an die Wand = where it is being fastened

A learner might expect something like a single verb meaning to screw onto, but here German expresses it as festschrauben + an die Wand.

Because an can take either accusative or dative, depending on meaning.

With two-way prepositions like an, German uses:

• accusative for direction / movement toward a place
• dative for location / position at a place

Here, the shelf is being attached to the wall, so the phrase is directional:

• an die Wand = onto/to the wall

That is why it uses accusative:

• die Wand

Compare:

• Das Regal hängt an der Wand. = The shelf is hanging on the wall.
  Here it is already in a location, so dative is used.

Das Regal is in the accusative case because it is the direct object of the verb.

The sentence is about what the father is fastening:

• Mein Vater = who does the action
• schraubt = does the action
• das Regal = what is being fastened

For das, nominative and accusative look the same, so you do not see a form change:

• nominative: das Regal
• accusative: das Regal

But its function here is accusative/direct object.

Because mein Vater is the subject of the sentence, so it is in the nominative case.

Also, with masculine nouns, mein in the nominative singular stays mein:

• mein Vater = my father

If it were accusative masculine, it would change:

• Ich sehe meinen Vater. = I see my father.

But here the father is doing the action, so it stays:

• Mein Vater

Not quite.

• schrauben by itself usually means to screw, to turn with a screw, or to work with screws
• festschrauben means to screw tight, to fasten securely with screws

So fest adds the idea of firmness or secure attachment.

That is why schraubt ... fest sounds more natural here than just schraubt if the point is that the shelf is being fixed firmly to the wall.

Yes, but the meaning changes slightly.

• Mein Vater schraubt das Regal an die Wand.
  This can still be understood, and many speakers would use it.
• Mein Vater schraubt das Regal an die Wand fest.
  This makes the idea of securely fastening more explicit.

So fest is not meaningless extra decoration. It emphasizes that the shelf is being fixed firmly in place.

The basic structure is:

• Mein Vater = subject
• schraubt = finite verb in position 2
• das Regal = object
• an die Wand = prepositional phrase
• fest = separable prefix at the end

So the pattern is:

Subject + conjugated verb + object + prepositional phrase + separable prefix

This is a very typical German main-clause pattern.

Because in a main clause, separable verbs split apart.

So:

• infinitive: festschrauben
• main clause: schraubt ... fest

Examples:

• Ich mache die Tür auf. from aufmachen
• Er ruft mich an. from anrufen
• Mein Vater schraubt das Regal fest. from festschrauben

But in infinitives or subordinate clauses, the verb stays together:

• Mein Vater will das Regal an die Wand festschrauben.
• ..., weil mein Vater das Regal an die Wand festschraubt.

In this sentence, it is closer to to the wall or onto the wall.

That is because the sentence describes the action of attaching the shelf so that it ends up on the wall. German expresses this with the directional phrase:

• an die Wand

If you wanted to describe the finished location, you would usually use:

• an der Wand = on the wall / attached to the wall

So a useful contrast is:

• Er schraubt das Regal an die Wand. = He fastens the shelf to the wall.
• Das Regal ist an der Wand. = The shelf is on the wall.

Yes, Regal is a neuter noun, and you can see that from the article:

• das Regal

That tells you the dictionary form is:

• das Regal

In this sentence, das happens to be the same in nominative and accusative singular for neuter nouns, so you must use sentence function to know the case. But the gender is clearly shown: Regal is neuter.

Yes. It sounds like normal, idiomatic German.

A native speaker would understand it as something like:

• My father is fastening the shelf to the wall with screws.

It is especially natural because:

• festschrauben is a common verb
• an die Wand is the normal way to express attachment to a wall
• the word order follows standard German main-clause rules

So this is a good example sentence for learning separable verbs and two-way prepositions.