Barnet fik nyt legetøj i fødselsdagsgave.

Because barnet is the definite form: the child.

• et barn = a child
• barnet = the child

Danish usually adds the definite ending directly to the noun instead of using a separate word like English the.

Here, barn is a neuter noun, so its definite singular ending is -et:

• et barn → barnet

Fik is the past tense of få.

• få = to get / to receive
• fik = got / received

It is an irregular verb, so the past tense is not formed by simply adding a regular ending.

A few useful forms:

• at få = to get
• får = get / gets
• fik = got
• har fået = have/has gotten, received

Because legetøj is a neuter noun, and adjectives in Danish change form to match the noun.

In the singular indefinite:

• en-word: ny
• et-word: nyt

Since it is:

• et legetøj = a toy / toy stuff

the adjective must be:

• nyt legetøj

Compare:

• en ny bog = a new book
• et nyt hus = a new house

Grammatically, legetøj is singular, but it often works like a mass noun in Danish, similar to English toy stuff or sometimes just toys in a general sense.

So nyt legetøj can mean:

• a new toy
• new toys
• new toy stuff

The exact meaning depends on context.

Danish does have a plural form, legetøjer, but in everyday language legetøj is very often used as an uncountable/general noun.

Because Danish often leaves out the article with mass nouns or when speaking in a general, non-counted way.

So:

• nyt legetøj = new toy(s), new toy stuff
• et nyt legetøj = a new toy, one specific new toy

Both are possible, but they are not exactly the same.

In your sentence, nyt legetøj sounds natural if the idea is simply that the child received new toys or some new toy item(s), without focusing on counting exactly one.

It is an idiomatic expression meaning as a birthday present.

A very common Danish pattern is:

• få noget i gave = receive something as a gift

So:

• Barnet fik nyt legetøj i fødselsdagsgave means that the new toy(s) were received as a birthday gift.

Here, i does not mean a literal physical in. It is part of a fixed expression.

Because Danish usually writes compound nouns as one word.

So:

• fødselsdag = birthday
• gave = gift

together become:

• fødselsdagsgave = birthday present

The -s- in the middle is a common linking element in Danish compounds.

This is very normal in Danish:

• sommerferie = summer holiday
• skolebog = school book
• fødselsdagsgave = birthday present

Because i fødselsdagsgave is a set expression, and in this kind of phrase Danish often uses the noun without an article.

So you get:

• i gave = as a gift
• i fødselsdagsgave = as a birthday gift

This is one of those places where Danish sounds natural without en or en/et article, even though English would often use a:

• as a birthday present

The sentence follows normal Danish main-clause word order:

• Barnet = subject
• fik = finite verb
• nyt legetøj = object
• i fødselsdagsgave = adverbial phrase / complement

So the pattern is basically:

Subject + Verb + Object + Extra information

Danish main clauses normally put the finite verb in the second position.

For example:

• Barnet fik nyt legetøj i fødselsdagsgave.
• I går fik barnet nyt legetøj i fødselsdagsgave.

In the second sentence, I går takes the first position, so fik still stays second.

Yes, it can be a little flexible.

Because legetøj is often used as a mass noun, nyt legetøj does not force a strict one toy reading. It can mean:

• one new toy
• several new toys
• new toy stuff in general

If you want to make one toy clearer, you could say:

• Barnet fik et nyt stykke legetøj i fødselsdagsgave. or sometimes simply:
• Barnet fik et nyt legetøj i fødselsdagsgave.

If you want to make several toys clearer, you could say:

• Barnet fik nye legetøjer i fødselsdagsgave.

Yes, som fødselsdagsgave is possible, but i fødselsdagsgave is very idiomatic after få.

Compare:

• Barnet fik nyt legetøj i fødselsdagsgave.
  Very natural with fik; this is the classic pattern.

• Barnet fik nyt legetøj som fødselsdagsgave.
  Also understandable and natural, but slightly more explicitly as a birthday present.

So both work, but i fødselsdagsgave fits especially well with the verb få.

You would replace fik with får:

• Barnet får nyt legetøj i fødselsdagsgave.

That means:

• The child gets/is getting new toy(s) as a birthday present

So:

• fik = got
• får = gets / is getting

This is a useful pair to learn:

• Barnet får ...
• Barnet fik ...