Genhusningen er nødvendig under udtørringen.

Because -en is the attached definite article for a singular common-gender noun.

• en genhusning = a rehousing / temporary relocation
• genhusningen = the rehousing / temporary relocation

In Danish, the definite article is usually added to the end of the noun instead of being a separate word before it.

That is normal in Danish. Unlike English, Danish usually marks definiteness with a suffix on the noun:

• en bil = a car
• bilen = the car

So:

• genhusningen = the rehousing
• udtørringen = the drying process

A separate word like den is usually used when there is an adjective before the noun:

• den nødvendige genhusning = the necessary rehousing

Because nødvendig agrees with the subject.

Here, the subject is genhusningen, which is a singular common-gender noun, so the adjective stays in its basic form:

• Genhusningen er nødvendig. = The rehousing is necessary.

If the subject were a singular neuter noun, you would use nødvendigt:

• Opholdet er nødvendigt. = The stay is necessary.

If the subject were plural, you would normally use nødvendige:

• Foranstaltningerne er nødvendige. = The measures are necessary.

For the same reason: it is also a singular definite noun.

• en udtørring = a drying-out / a drying process
• udtørringen = the drying-out / the drying process

In this sentence, it refers to a specific drying process already known from the context, not just drying in general.

Here, under means during, not physically underneath.

So:

• under udtørringen = during the drying process

Danish under can be used both literally and temporally:

• under bordet = under the table
• under mødet = during the meeting

In this sentence, it is clearly temporal.

They are nouns made from verbs, using the ending -ing.

• at genhuse = to rehouse / relocate

• genhusning = rehousing / relocation

• at udtørre or at tørre ud = to dry out
• udtørring = drying out / drying process

This -ing ending often makes an action noun in Danish. It can look similar to English -ing, but it is a noun here, not a verb form.

Both are singular.

You can tell because:

• they each have the singular definite ending -en
• the adjective nødvendig is singular agreement

If they were plural, they would look different:

• genhusninger = rehousings / relocations

• genhusningerne = the rehousings / relocations

• udtørringer = drying processes
• udtørringerne = the drying processes

The structure is:

• Genhusningen = subject
• er = verb
• nødvendig = predicative adjective / subject complement
• under udtørringen = time phrase

So the pattern is:

Subject + verb + complement + adverbial

This is a very normal Danish main-clause pattern.

It also follows the Danish V2 rule: in a main clause, the finite verb normally comes in the second position. Since the subject comes first here, er comes next.

You could also front the time phrase:

• Under udtørringen er genhusningen nødvendig.

That is still correct, and er still stays in second position.

You could, but it would sound more general, abstract, or headline-like.

The original sentence with definite forms:

• Genhusningen er nødvendig under udtørringen.

suggests a specific situation:

• a particular rehousing
• a particular drying process

Without the definite endings:

• Genhusning er nødvendig under udtørring.

the sentence sounds more like a general statement such as Rehousing is necessary during drying-out.

So the definite forms are used because the speaker is talking about a known, concrete case.

Yes. It is an adjective used predicatively.

That means it comes after a verb like er and describes the subject:

• Genhusningen er nødvendig.

Compare:

• den nødvendige genhusning = the necessary rehousing
  
  • adjective before the noun: attributive
• Genhusningen er nødvendig.
  
  • adjective after er: predicative

English works the same way:

• the necessary relocation
• the relocation is necessary