Hun er generøs med sin tid, og tilfældigvis bor hun desuden lige ved siden af skolen.

Because sin/sit/sine is the reflexive possessive in Danish. It refers back to the subject of the same clause.

Here, the subject is hun, so sin tid means her own time.

• Hun er generøs med sin tid = she is generous with her own time
• Hun er generøs med hendes tid would normally mean she is generous with another woman’s time

This is one of the most important Danish grammar points for English speakers, because English uses her for both meanings, but Danish makes a clear distinction.

It is a very natural expression, and it works much like English generous with her time.

Structure:

• generøs = generous
• med = with
• sin tid = her time / her own time

So the pattern is:

• at være generøs med noget = to be generous with something

Examples:

• Han er generøs med sine penge = He is generous with his money
• Hun er generøs med sin hjælp = She is generous with her help

So yes, it is both grammatical and idiomatic.

Because Danish main clauses follow the V2 rule: the finite verb must come in second position.

In the second clause, tilfældigvis has been placed first for emphasis. Once that happens, the verb must come next:

• Tilfældigvis bor hun ...

Breakdown:

1. tilfældigvis = first element
2. bor = finite verb in second position
3. hun = subject after the verb

This is very normal Danish word order.

Compare:

• Hun bor lige ved siden af skolen = neutral order
• Tilfældigvis bor hun lige ved siden af skolen = with tilfældigvis fronted

Tilfældigvis means by chance, coincidentally, or as it happens.

In this sentence, it comments on the whole second clause, not just one word. It tells us that the fact that she lives next to the school is a coincidence.

So:

• tilfældigvis = coincidentally / by chance

It is often used as a sentence adverb and can appear in different positions, for example:

• Hun bor tilfældigvis lige ved siden af skolen
• Tilfældigvis bor hun lige ved siden af skolen

Both are possible, but fronting it gives it more emphasis.

Desuden means besides, furthermore, or in addition.

It adds another point:

• first point: she is generous with her time
• second point: she also happens to live right next to the school

So the two adverbs do different jobs:

• tilfældigvis = this is a coincidence
• desuden = this is an additional fact

You could think of the second clause as meaning something like:

• and, additionally, she happens to live right next to the school

Because og links two main clauses, and each main clause still follows normal Danish main-clause word order.

First clause:

• Hun er generøs med sin tid

Second clause:

• tilfældigvis bor hun desuden lige ved siden af skolen

Even after og, if the second clause starts with something other than the subject, Danish still uses V2 word order.

So:

• og hun bor ... = possible if the subject comes first
• og tilfældigvis bor hun ... = also correct, because tilfældigvis is first and the verb must be second

Ved siden af means next to or beside.

Adding lige makes it stronger:

• ved siden af = next to
• lige ved siden af = right next to / immediately next to

So lige adds emphasis and makes the location feel very close.

Examples:

• Hun sidder ved siden af mig = She is sitting next to me
• Hun bor lige ved siden af skolen = She lives right next to the school

Skolen is the definite form of skole:

• en skole = a school
• skolen = the school

Danish usually expresses definiteness by adding the article to the end of the noun.

Here, skolen suggests a specific school that is known from the context.

So:

• ved siden af skolen = next to the school

This is very common in Danish:

• bilen = the car
• huset = the house
• skolen = the school

Yes:

• tilfældigvis = coincidentally / by chance
• bor = lives
• hun = she
• desuden = besides / in addition
• lige = right / directly
• ved siden af = next to
• skolen = the school

So the clause structure is:

• fronted adverb: tilfældigvis
• finite verb: bor
• subject: hun
• sentence adverb: desuden
• place expression: lige ved siden af skolen

That order is very natural in Danish.

Yes. Bor is the present tense of bo, which means to live / reside.

• at bo = to live
• hun bor = she lives

A useful point for English speakers: Danish often uses the simple present where English might use either:

• she lives
• she is living

So hun bor lige ved siden af skolen most naturally means she lives right next to the school.

Yes, but it can sound a little more formal or polished than også.

Roughly:

• også = also
• desuden = besides / moreover / in addition

So desuden is perfectly correct and natural, especially in careful speech or writing. In casual everyday speech, some people might more often say også, depending on the exact nuance they want.

Here, desuden works well because it adds an extra supporting fact, not just a simple also.