Der Mann wirkt freundlich, aber ich vertraue dem nicht.

Not quite.

wirken here means to seem, to come across, or to appear. So Der Mann wirkt freundlich means the man gives the impression of being friendly.

That is a little different from Der Mann ist freundlich, which states more directly that he actually is friendly.

So:

• Der Mann ist freundlich = he is friendly
• Der Mann wirkt freundlich = he seems friendly / comes across as friendly

The sentence suggests some doubt: he appears nice, but the speaker is not convinced.

Because freundlich is a predicate adjective here.

After verbs like:

• sein = to be
• werden = to become
• bleiben = to remain
• wirken = to seem/appear

the adjective usually does not take an ending.

So:

• Der Mann ist freundlich
• Der Mann wirkt freundlich
• Der Mann bleibt freundlich

But when the adjective comes before a noun, it does take an ending:

• ein freundlicher Mann
• der freundliche Mann

That is the difference.

Because vertrauen takes the dative case, not the accusative.

So the pattern is:

• jemandem vertrauen = to trust someone

With der Mann, the forms are:

• nominative: der Mann
• accusative: den Mann
• dative: dem Mann

So if you want to say I trust the man, it would be:

• Ich vertraue dem Mann.

That is why the sentence uses dem, not den.

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

• ich vertraue ihm nicht = I don’t trust him
• ich vertraue dem nicht = I don’t trust him / I don’t trust that guy

Here dem is acting like a demonstrative pronoun. It can sound a bit more pointed, a bit more distancing, and sometimes a little colloquial or dismissive.

So:

• ihm is more neutral
• dem can feel more like that one / that guy / him

In this sentence, dem fits well because the speaker has just mentioned Der Mann and is now referring back to him with a slightly skeptical attitude.

Because nicht is usually placed near what it negates, and in this sentence it negates the whole idea of trusting him.

• Ich vertraue dem nicht. = I do not trust him.

In simple main clauses, nicht often comes quite late, especially when it negates the verb phrase rather than a single word.

Compare:

• Ich vertraue dem nicht. = I don’t trust him.
• Nicht ich vertraue dem. = It’s not me who trusts him.
  (This changes the emphasis.)
• Ich vertraue nicht dem Mann, sondern der Frau. = I trust not the man, but the woman.
  (Here nicht contrasts the object.)

So the final position is normal here.

Because aber is a coordinating conjunction.

Coordinating conjunctions such as:

• aber = but
• und = and
• oder = or
• denn = for/because

do not push the verb to the end.

So you get:

• Der Mann wirkt freundlich, aber ich vertraue dem nicht.

The second clause still has normal main-clause word order, with the verb in second position:

• ich vertraue ...

Compare that with a subordinating conjunction like weil:

• ..., weil ich dem nicht vertraue.

There the verb goes to the end.

Because the same noun is playing different grammatical roles in the two clauses.

In the first clause:

• Der Mann wirkt freundlich

Der Mann is the subject, so it is in the nominative case.

In the second clause:

• ich vertraue dem nicht

the man is the object of vertrauen, and since vertrauen takes the dative, the form becomes dem.

So the change is:

• der for nominative
• dem for dative

This is a very common feature of German: articles and pronouns change form depending on case.

Yes, absolutely.

That version is very natural and probably the more neutral choice in many situations.

• Der Mann wirkt freundlich, aber ich vertraue ihm nicht.

Using ihm simply means him, without the extra distancing flavor of dem.

So both are possible:

• ... aber ich vertraue ihm nicht. = neutral
• ... aber ich vertraue dem nicht. = a bit more pointed/distancing

By form alone, dem can be either:

• masculine dative singular
• neuter dative singular

But in this sentence, the reference is clearly to Der Mann, which is masculine.

So here dem is masculine dative.

The context tells you what it refers to.

Because the subject is ich.

The verb is vertrauen, and in the present tense:

• ich vertraue
• du vertraust
• er/sie/es vertraut
• wir vertrauen
• ihr vertraut
• sie/Sie vertrauen

So:

• ich vertraue = I trust
• er vertraut = he trusts

That is why the sentence has ich vertraue, not ich vertraut.