Hendes forsikring er dyr, men hun føler sig mere sikker med den.

Because Danish uses sin/sit/sine only when the possessor is also the subject of the same clause.

Here, the clause is Hendes forsikring er dyr. The subject is forsikring (the insurance), not hun. So Danish uses hendes.

Compare:

• Hun elsker sin forsikring. = She loves her own insurance.
• Hendes forsikring er dyr. = Her insurance is expensive.

So hendes is correct here.

Because dyr is being used after er, so it is a predicative adjective.

In Danish, adjectives often look different depending on whether they come:

• before a noun: en dyr forsikring
• after a verb like er: Forsikringen er dyr

Here forsikring is singular and common gender, so dyr is the correct form.

Sig is a reflexive pronoun. In Danish, føle sig is the normal way to say feel when talking about how someone feels.

So:

• hun føler sig mere sikker = she feels safer / more secure

English does not usually keep the reflexive word here, but Danish does.

Similar examples:

• Jeg føler mig træt. = I feel tired.
• Han føler sig bedre. = He feels better.

Normally, no. With this meaning, Danish usually needs the reflexive pronoun:

• hun føler sig mere sikker

Without sig, the sentence sounds wrong or incomplete to most speakers.

Yes, sikrere is also possible and is the regular one-word comparative of sikker.

So both of these can mean roughly the same thing here:

• mere sikker
• sikrere

A learner will probably hear sikrere quite often, but mere sikker is also understandable and natural enough in context.

Because den refers back to forsikring, and forsikring is a common-gender noun:

• en forsikring

In Danish:

• den = it for common gender singular nouns
• det = it for neuter singular nouns

So med den means with it, where it refers to the insurance.

Here med den means something like:

• with it
• by having it
• with that insurance in place

It does not mean she is physically holding the insurance. It means that having the insurance makes her feel more secure.

Because men starts a new main clause. After men, Danish normally keeps ordinary main-clause word order:

• men hun føler sig mere sikker med den

You get inversion only when some other element is placed first. For example:

• I dag føler hun sig mere sikker.
• Med den føler hun sig mere sikker.

But after men alone, hun føler is the normal order.

Because men is connecting two clauses:

• Hendes forsikring er dyr
• hun føler sig mere sikker med den

In standard Danish punctuation, a comma is normally written before men when it joins clauses like this.

Here it means safe / secure, not certain.

Danish sikker can mean sure/certain in other contexts, but in this sentence it is about a feeling of security.

Here forsikring means insurance, most naturally her insurance coverage or insurance policy.

So Hendes forsikring er dyr means her insurance is expensive. It is not talking about a promise or assurance; it is talking about insurance in the financial/practical sense.