Jeg har tillid til hende, fordi hun er opmærksom på små fejl og aldrig virker usikker.

At have tillid til nogen is a very common Danish expression meaning to have trust in someone / to trust someone.

It is built like this:

• jeg har = I have
• tillid = trust, confidence
• til hende = in her / toward her

So Danish often expresses this idea with a noun phrase rather than a single verb.

A very close alternative is jeg stoler på hende, which also means I trust her.

Because after a preposition such as til, Danish uses the object form of the pronoun.

So:

• hun = she
• hende = her

That is the same kind of difference as English she vs her.

Examples:

• Hun er her. = She is here.
• Jeg taler med hende. = I am speaking with her.
• Jeg har tillid til hende. = I trust her.

Because tillid normally goes with the preposition til. It is just the standard pattern:

• tillid til nogen/noget = trust in someone/something

In the same way, English has patterns that must simply be learned together, such as belief in or confidence in.

So it is best to learn tillid til as one unit.

Because opmærksom normally takes the preposition på.

• at være opmærksom på noget = to be aware of something / to pay attention to something

So:

• hun er opmærksom på små fejl = she notices / pays attention to small errors

Again, this is best learned as a fixed combination: opmærksom på.

This is about word order in a subordinate clause.

After fordi, Danish normally uses subordinate-clause word order, and sentence adverbs such as aldrig, ikke, ofte, etc. come before the finite verb.

So:

• main clause: Hun virker aldrig usikker.
• subordinate clause: ... fordi hun aldrig virker usikker.

In your sentence, the part after fordi is subordinate, so aldrig comes before virker.

Because the same subject, hun, is shared by both parts:

• hun er opmærksom på små fejl
• (hun) aldrig virker usikker

When two verbs or verb phrases have the same subject, Danish often leaves the second subject out if it is already clear.

So the sentence is understood as:

• ... fordi hun er opmærksom på små fejl og (hun) aldrig virker usikker.

That omission sounds natural and avoids repetition.

Because they are predicate adjectives after verbs like er and virker.

With a singular common-gender subject such as hun, the adjective usually appears in its basic form:

• hun er opmærksom
• hun virker usikker

You would see other forms in other contexts, for example:

• det er usikkert = it is uncertain
• de er usikre = they are uncertain

So here the basic form is the correct one.

Because små fejl refers to small errors in a general sense, not to a specific set already identified.

• små fejl = small errors
• de små fejl = the small errors

So in this sentence, the idea is that she is attentive to small mistakes in general.

Also, note that små is the plural form used with fejl here.

Here virker does not mean works/functions. It means seems, appears, or comes across as.

So:

• hun virker usikker = she seems uncertain / she comes across as insecure

This is a very common use of virke in Danish.

A similar phrase is:

• hun ser usikker ud = she looks uncertain

But virker usikker is a bit broader: it is about the overall impression she gives, not only her visual appearance.

Yes, absolutely. Jeg stoler på hende is very natural.

Both can mean I trust her, but there can be a slight nuance:

• jeg har tillid til hende can sound a bit more like I have confidence in her
• jeg stoler på hende can sound a bit more direct and everyday

In many situations, either one works well.