Ég hef fullt traust á vinkonu minni.

Because Icelandic can express this idea with the noun traust and the verb að hafa:

• að hafa traust á einhverjum = to have trust/confidence in someone

So the sentence is built like I have full trust in my friend.

There is also a very common verb-based way to say this:

• Ég treysti vinkonu minni. = I trust my friend.

The version with hef fullt traust sounds a bit more explicit and emphasizes the strength of that trust.

Fullt traust means full trust, complete trust, or complete confidence.

The word traust is a neuter singular noun, so the adjective has to match it in gender, number, and case. That is why you get:

• fullt
  • traust

Here, fullt is the neuter singular form of fullur.

So this is not an adverb like fully. It is an adjective agreeing with the noun traust.

It is the accusative singular, because it is the direct object of hef.

• Ég = subject
• hef = verb
• fullt traust = direct object

A useful thing to know is that for many neuter nouns in Icelandic, the nominative and accusative singular look the same. So traust looks the same in both cases.

Not literally. In this sentence, á is part of a fixed expression:

• að hafa traust á einhverjum = to have trust/confidence in someone

So even though á often means on, here you should learn the whole phrase as a unit.

This is very common in Icelandic: a noun or verb often goes with a particular preposition, and the result does not translate word-for-word into English.

Because the preposition á in this expression takes the dative, so both the noun and the possessive have to change form.

Dictionary forms:

• vinkona = female friend
• minn = my

But in this sentence, they appear in the dative singular feminine:

• vinkonu
• minni

So:

• vinkona mín = nominative, my female friend
• vinkonu minni = dative, to/in/on my female friend depending on the phrase

A very helpful clue is minni. That form clearly shows the dative feminine singular.

Yes. Vinkona specifically refers to a female friend.

Related words:

• vinur = a male friend, or sometimes just friend in a more general sense depending on context
• vinkona = a female friend

So by choosing vinkona, the sentence tells you that the friend is female.

Because in Icelandic, possessives like minn, þinn, and sinn very often come after the noun:

• vinur minn
• vinkona mín
• vinkonu minni

That is the normal, neutral pattern.

You can put the possessive before the noun for emphasis or contrast, but that is less neutral:

• mín vinkona

So in your sentence, vinkonu minni is the ordinary, natural word order.

Yes, absolutely.

That is a very natural alternative and means I trust my friend.

The difference is mainly in structure and emphasis:

• Ég treysti vinkonu minni. = I trust my friend.
• Ég hef fullt traust á vinkonu minni. = I have full/complete trust in my friend.

The second version stresses the degree of trust a bit more.

Also note the grammar difference:

• treysta einhverjum takes a direct object in the dative
• hafa traust á einhverjum uses the noun traust plus the preposition á

Yes. This is a very normal, neutral Icelandic sentence pattern:

• Ég = subject
• hef = verb
• fullt traust = object
• á vinkonu minni = prepositional phrase

So the structure is basically:

• Subject + Verb + Object + Prepositional phrase

Icelandic word order can be more flexible than English, especially for emphasis, but this sentence is the straightforward, unmarked way to say it.