Hún trúir mér ekki alveg, en ég segi henni satt samt.

Because the verb trúa takes the dative when it means believe someone.

• trúir mér = believes me
• mér is the dative form of ég
• mig is accusative, and that is not the form this verb wants here

So this is a case pattern you need to learn with the verb:

• ég trúi honum
• hún trúir mér
• við trúum þeim

They mean different things:

• trúa einhverjum = believe someone
• trúa á einhvern = believe in someone

So:

• Hún trúir mér ekki alveg = She does not quite believe me
• Hún trúir á mig = She believes in me

That is a very important distinction in Icelandic.

Because segja usually works like segja einhverjum eitthvað = tell someone something.

That means:

• the person is in the dative
• the thing said is the content

So here:

• henni = to her / her in dative
• satt = the thing being said

Compare:

• Ég segi henni söguna = I tell her the story
• Ég segi henni satt = I tell her the truth

Satt is the neuter singular form of the adjective sannur = true.

In the fixed expression segja satt, Icelandic uses this neuter form to mean something like:

• tell the truth
• speak truthfully

So even though satt is originally an adjective form, the whole phrase segja satt is a common idiomatic expression.

You can also say:

• segja sannleikann = tell the truth

But segja satt is shorter and very common.

It means not completely, not entirely, or more naturally in English, not quite.

So:

• trúir mér ekki alveg = does not quite believe me

The word alveg softens the negation a little. It is not a full, flat she doesn't believe me. It is more like:

• she is not fully convinced
• she does not completely believe me

Because the meaning changes.

• ekki alveg = not quite / not completely
• alveg ekki = not at all / absolutely not

So in this sentence:

• Hún trúir mér ekki alveg = She does not quite believe me

If you said:

• Hún trúir mér alveg ekki

that would be much stronger: She absolutely does not believe me.

There are two useful things going on here.

1. Icelandic main clauses are verb-second So when the subject comes first, the finite verb usually comes second:

• Hún trúir ...
• ég segi ...

2. Short pronouns often come early Unstressed pronouns like mér and henni usually appear before adverbs like ekki.

So:

• Hún trúir mér ekki alveg sounds natural

A version like Hún trúir ekki alveg mér would sound unusual unless mér were being strongly contrasted.

Here samt means anyway, still, or all the same.

So:

• en ég segi henni satt samt = but I tell her the truth anyway

It shows contrast: even though she does not fully believe me, I still tell her the truth.

Yes, samt can appear in different places, depending on style and emphasis.

This sentence has:

• ég segi henni satt samt

That end position gives samt a slightly emphatic anyway feeling.

You could also hear:

• ég segi henni samt satt

Both are possible. The meaning is basically the same, but the rhythm and emphasis shift a little.

Because they come from different verbs and agree with different subjects.

• trúir comes from trúa
• segi comes from segja

In this sentence:

• Hún trúir = she believes
  third person singular present

• Ég segi = I say / I tell
  first person singular present

A quick comparison:

• ég trúi / hún trúir
• ég segi / hún segir

So the forms differ both because of the subject and because the verbs themselves are conjugated differently.

Yes, very often it does.

English makes a stronger distinction between say and tell, but Icelandic often uses segja for both.

• Ég segi = I say
• Ég segi henni satt = I tell her the truth
• Ég segi að... = I say that...

So the exact English translation depends on the structure around segja.

Yes:

• Hún = she
• trúir = believes
• mér = me, dative
• ekki alveg = not quite / not completely
• en = but
• ég = I
• segi = say / tell
• henni = her, dative
• satt = true / truthfully / the truth
• samt = anyway / still

A very literal structural gloss would be:

• She believes me not completely, but I say her true anyway

That is not good English, of course, but it shows how the Icelandic sentence is built.