De spijker is krom, dus Anna pakt liever een nieuwe en gebruikt de hamer opnieuw.

Because spijker is a de-word in Dutch. Nouns in Dutch have grammatical gender, and you usually have to learn the article with the noun:

• de spijker = the nail
• plural: de spijkers

There is no simple rule that lets you always predict whether a noun takes de or het, so learners usually memorize the noun together with its article.

Because krom comes after the verb is, so it is a predicative adjective. Predicative adjectives normally do not take -e in Dutch.

• De spijker is krom. = The nail is bent.
• De kromme spijker = the bent nail

So:

• after zijn / worden / blijven / lijken, etc. → usually krom
• before a noun → usually kromme

Dus means so, therefore, or as a result.

In this sentence:

• De spijker is krom, dus Anna pakt liever een nieuwe...
• The nail is bent, so Anna would rather take a new one...

It connects the first idea to the consequence that follows.

Here dus is linking two main clauses in a way similar to English so, and Dutch commonly keeps normal main-clause order after it:

• ..., dus Anna pakt...

That is very natural.

Learners often expect inversion after any word placed early in a clause, but dus is special in actual usage. Compare:

• ..., dus Anna pakt liever een nieuwe. = natural
• Daarom pakt Anna liever een nieuwe. = with daarom, inversion is required

So a useful shortcut is:

• dus often behaves like so
• daarom behaves more like therefore, with inversion

Liever means rather or preferably.

So:

• Anna pakt liever een nieuwe.
• Anna would rather take a new one.

Dutch often uses liever where English might use would rather or prefer.

Some useful patterns:

• Ik drink liever thee. = I’d rather drink tea.
• Liever dit dan dat. = Rather this than that.
• Ze pakt liever een nieuwe spijker. = She prefers to take a new nail.

Because Dutch, like English, can leave out a noun when it is obvious from the context.

• een nieuwe here means a new one
• the missing noun is spijker

So:

• Anna pakt liever een nieuwe
  literally: Anna would rather take a new one

You could also say:

• Anna pakt liever een nieuwe spijker

Both are correct. The version without the noun sounds natural because spijker has already been mentioned.

Because the omitted noun is spijker, and spijker is a de-word.

When an adjective comes before a singular de-word, it normally takes -e:

• een nieuwe spijker
• de nieuwe spijker

When the noun is omitted, the adjective keeps the same form:

• een nieuwe = a new one

If the hidden noun were a singular het-word, you might get nieuw instead:

• een nieuw huis = a new house

So the -e in nieuwe tells you something about the type of noun being left out.

Because the sentence is referring to a specific hammer, or at least to the hammer that is relevant in the situation.

• de hamer = the hammer
• een hamer = a hammer, any hammer

Using de hamer suggests the hammer is already understood from the context. That is very natural when talking about a familiar tool in a scene like this.

If you said een hamer, it would sound more like she is taking some hammer, not a particular one.

Yes, weer could also work here, but the nuance is slightly different.

weer

Usually means again in a general sense.

• Ze gebruikt de hamer weer. = She uses the hammer again.

opnieuw

Often means again / anew / once more, and can sound a bit more deliberate, like repeating the action or starting over.

• Ze gebruikt de hamer opnieuw.

In this sentence, opnieuw fits well because the first attempt with the bent nail did not work out, so she is trying again with a new nail.

So:

• weer = common everyday again
• opnieuw = again, often with a sense of repetition or retrying

Because that is a very natural place for an adverb in Dutch.

The basic structure here is:

• Anna = subject
• gebruikt = finite verb
• de hamer = object
• opnieuw = adverb

So:

• ... en gebruikt de hamer opnieuw.

Putting opnieuw at the end sounds normal and smooth.

You may also see other word orders in Dutch depending on emphasis, but sentence-final placement is very common for adverbs like this.

Yes, neemt is possible, but pakt is very natural here.

pakken

Usually means something like:

• grab
• pick up
• take in a physical, concrete way

nemen

Is more general:

• take

Because Anna is physically taking a new nail, pakt sounds very idiomatic.

Compare:

• Anna pakt een nieuwe spijker. = Anna picks up / takes a new nail.
• Anna neemt een nieuwe spijker. = also possible, but a little less physical-sounding

So pakt is a good, natural choice in this context.

Because the subject is the same for both verbs, and Dutch does not need to repeat it.

• Anna pakt liever een nieuwe en gebruikt de hamer opnieuw.

This works just like English:

• Anna takes a new one and uses the hammer again.

You could repeat the subject for emphasis or clarity:

• ..., en Anna gebruikt de hamer opnieuw.

But normally that would sound unnecessary here.

Yes, exactly. Een nieuwe is short for een nieuwe spijker.

This kind of omission is very common in Dutch when the noun is already clear:

• Ik neem de rode. = I’ll take the red one.
• Wil je een grote of een kleine? = Do you want a big one or a small one?
• Anna pakt liever een nieuwe. = Anna would rather take a new one.

So this is a very useful pattern to recognize and use.