Sæt bogen på plads, når du er færdig.

Because Sæt is the imperative form of at sætte (to put / to place).

Danish uses the imperative for commands, instructions, and requests, just like English uses Put ...

• at sætte = to put
• sætter = puts / is putting
• sæt = put!

So:

• Sæt bogen på plads = Put the book away / back in its place

In commands, Danish usually leaves out the subject du, just like English often leaves out you:

• (Du,) sæt bogen på plads. = (You,) put the book away.

Because bogen means the book, while bog means a book or just book in a general sense.

Danish usually adds the definite article to the end of the noun:

• en bog = a book
• bogen = the book

So in this sentence, bogen is a specific book that the listener is expected to put away.

This is a normal feature of Danish grammar. Unlike English, Danish often puts the as a suffix on the noun.

For common-gender nouns:

• en stol = a chair
• stolen = the chair

For neuter nouns:

• et bord = a table
• bordet = the table

Since bog is a common-gender noun (en bog), the definite form becomes bogen.

På plads is a very common expression meaning something like:

• in its proper place
• back where it belongs
• put away

So Sæt bogen på plads is not just literally put the book on place. It means put the book back where it belongs.

You will see på plads in many everyday situations:

• Læg tøjet på plads. = Put the clothes away.
• Sæt glassene på plads. = Put the glasses back in their place.

It is basically an idiomatic expression.

Because på plads often works as a fixed expression on its own.

Danish does allow phrases like:

• på sin plads = in its place

But in many everyday instructions, people simply say på plads, especially when it is obvious that the object should go back to where it belongs.

So:

• Sæt bogen på plads sounds natural and idiomatic.
• Sæt bogen på sin plads is possible, but it can sound more explicit or slightly heavier.

Når means when, and it is used here because the sentence refers to the time when something is completed.

• Sæt bogen på plads, når du er færdig.
• Put the book away when you are finished.

A useful comparison is:

• når = when
• hvis = if

In this sentence, the speaker expects the action to happen, so when is more natural than if.

Using hvis would change the meaning:

• ..., hvis du er færdig = ..., if you are finished

That sounds less certain.

Because når du er færdig is a subordinate clause, and Danish word order changes in subordinate clauses.

In a main clause, Danish normally has verb-second word order:

• Du er færdig. = You are finished.

But in a subordinate clause introduced by når, the subject usually comes before the verb in the normal way:

• når du er færdig

So:

• Hvornår er du færdig? = When are you finished?
  (question word order)
• når du er færdig = when you are finished
  (subordinate clause word order)

Because færdig agrees with the subject.

Here the subject is du, referring to one person, so the adjective stays in its basic form:

• du er færdig = you are finished

Compare:

• det er færdigt = it is finished
  (neuter singular)
• vi er færdige = we are finished
  (plural)

So færdig is correct because the sentence is talking to one person.

Here færdig means finished / done.

So når du er færdig means:

• when you are finished
• when you are done

It often refers to finishing an activity:

• Jeg er færdig med lektierne. = I’m finished with the homework.
• Er du færdig? = Are you done?

In this sentence, it means when you have finished whatever you are doing.

Because når du er færdig is a subordinate clause, and Danish often places a comma before subordinate clauses.

So the comma helps show the structure:

• main clause: Sæt bogen på plads
• subordinate clause: når du er færdig

In modern Danish, comma rules can be taught slightly differently depending on whether someone uses traditional comma or new comma, but in ordinary written Danish, a comma here is very common and expected.

Yes, absolutely. That is also correct Danish.

Both versions are natural:

• Sæt bogen på plads, når du er færdig.
• Når du er færdig, sæt bogen på plads.

The difference is mainly one of emphasis and flow.

• Starting with Sæt bogen på plads puts the command first.
• Starting with Når du er færdig puts the timing first.

Both mean the same thing.

Yes, very often you could.

Both sætte and lægge can be used for putting something somewhere, but they are not always identical.

Very roughly:

• sætte is often used for placing something in an upright position or just putting something somewhere
• lægge is often used for laying something down

In everyday speech, though, with objects like books, both can sound natural depending on context:

• Sæt bogen på plads
• Læg bogen på plads

The first sentence is very idiomatic as a general instruction to put the book back where it belongs.