Vores nye lærer er både grundig og pålidelig, så jeg føler mig tryg i hendes timer.

Because after a possessive like vores (our), Danish adjectives usually take the -e ending.

• vores nye lærer = our new teacher
• Compare:
  • en ny lærer = a new teacher
  • den nye lærer = the new teacher
  • vores nye lærer = our new teacher

So nye is the expected form here.

They are predicate adjectives, not adjectives placed directly before the noun.

In this sentence:

• Vores nye lærer = noun phrase
• er både grundig og pålidelig = is both thorough and reliable

After er (is), Danish normally uses the basic adjective form:

• Hun er grundig
• Han er pålidelig
• Læreren er tryg

You would more often see -e when the adjective comes before a noun in a definite context:

• den grundige lærer
• den pålidelige lærer

Både ... og means both ... and.

So:

• både grundig og pålidelig = both thorough and reliable

It is a fixed pairing in Danish, just like in English:

• Hun er både sød og klog = She is both kind and smart
• Jeg taler både dansk og engelsk = I speak both Danish and English

If you use både, you normally also use og later in the phrase.

Grundig usually means thorough, careful, or systematic.

About a teacher, it suggests that the teacher:

• explains things properly
• pays attention to details
• does not rush through material

It is usually positive. In this sentence, grundig is best understood as thorough.

Yes, pålidelig is very close to reliable, and sometimes trustworthy depending on context.

About a teacher, it suggests someone who:

• is dependable
• does what they say
• can be trusted
• is consistent

So grundig og pålidelig gives a very positive description: the teacher is both careful and dependable.

Because føle in this kind of sentence usually takes an object pronoun: mig, dig, sig, etc.

So Danish says:

• jeg føler mig tryg = I feel safe
• hun føler sig glad = she feels happy
• vi føler os hjemme = we feel at home

This is a very common Danish pattern. English does not use the extra pronoun here, but Danish does.

They can both relate to safe, but they are not the same.

Tryg is about feeling emotionally safe, comfortable, secure, and at ease.

• Jeg føler mig tryg = I feel safe / secure / comfortable

Sikker often means:

1. safe in a practical sense
   
   • Det er sikkert = It is safe
2. certain / sure
   
   • Jeg er sikker = I am sure

In this sentence, tryg is the natural choice because the speaker is talking about feeling comfortable and secure in the teacher’s classes.

Because hendes refers to the teacher, who is not the subject of this clause.

The clause is:

• jeg føler mig tryg i hendes timer

Here, the subject is jeg (I).
The possessive refers to lærer (teacher), not to jeg.

In Danish, sin/sit/sine is used when the possessor is the same as the subject of the clause.

So:

• Jeg vasker mine hænder = I wash my hands
• Hun elsker sin lærer = She loves her own teacher

But here, the subject is I, and the possessor is her, so:

• i hendes timer = in her classes

Not in this sentence. Here timer means lessons or classes.

Danish time can mean:

• hour
• lesson/class period

So:

• to timer can mean two hours
• i hendes timer means in her classes / during her lessons

This is a common point of confusion for English speakers.

Yes, literally it is in her lessons/classes, and that is a natural Danish way to express the idea.

So:

• jeg føler mig tryg i hendes timer

means something like:

• I feel safe in her classes
• I feel comfortable during her lessons

The preposition i is very natural here, even if English might also use during in some contexts.

Because så here is functioning like the coordinating conjunction so, linking two main clauses.

• Vores nye lærer er både grundig og pålidelig
• så jeg føler mig tryg i hendes timer

After a coordinating conjunction like this, Danish usually keeps normal main-clause order:

• så jeg føler ...

If another element were placed first inside the clause, then inversion could happen, but here the subject jeg comes first in the clause.

So the structure is very natural as written.

The noun lærer itself is not grammatically feminine in the way nouns are in some languages. Danish mainly has common gender and neuter.

• en lærer = a teacher (common gender)

But hendes is used because the sentence is talking about a female teacher in real life, not because lærer is a feminine noun.

So:

• hendes timer = her classes if the teacher is female
• hans timer = his classes if the teacher is male

A natural English version could be:

• Our new teacher is both thorough and reliable, so I feel safe in her classes.

Depending on context, you might also say:

• Our new teacher is both thorough and dependable, so I feel comfortable in her classes.
• Our new teacher is very thorough and reliable, so I feel at ease in her lessons.

The key point is that tryg often includes a sense of emotional comfort, not just physical safety.