Samtalið varð vandræðalegt þegar hann viðurkenndi að þetta hefði verið lygi.

Because -ið is the attached definite article for a neuter singular noun here.

• samtal = conversation
• samtalið = the conversation

Icelandic usually puts the on the end of the noun instead of using a separate word like English does.

Because verða means to become / to get / to turn into, while vera means to be.

So:

• Samtalið var vandræðalegt = The conversation was awkward
• Samtalið varð vandræðalegt = The conversation became awkward

In this sentence, the idea is that the conversation changed and turned awkward at that moment.

Because the adjective has to agree with samtalið.

• samtal is neuter
• samtalið is neuter singular
• so the adjective must also be neuter singular: vandræðalegt

This is true even when the adjective comes after the verb, as in varð vandræðalegt.

It means awkward or embarrassing in the sense of causing discomfort or social tension.

It is related to vandræði, which means trouble, difficulty, or often embarrassment / awkwardness depending on context.

So vandræðalegt is basically awkward/embarrassing as an adjective.

Because this is a subordinate clause introduced by þegar (when).

In main clauses, Icelandic often follows a verb-second pattern, but subordinate clauses usually do not. So after þegar, the normal order is:

• þegar hann viðurkenndi ...
• literally: when he admitted ...

That is the expected word order here.

The dictionary form is viðurkenna, meaning to admit, to acknowledge, or to recognize, depending on context.

Here:

• hann viðurkenndi = he admitted

This is the past tense form.

Here að means that and introduces the content of what he admitted.

So:

• hann viðurkenndi að ... = he admitted that ...

This is a very common pattern in Icelandic, just like English uses that after verbs such as say, think, know, admit, and so on.

Yes, word-for-word it can look like would have been, but here it functions as a subjunctive perfect form and is often translated naturally as had been.

• hefði = subjunctive form of hafa
• verið = past participle of vera

So:

• að þetta hefði verið lygi = that this had been a lie

Why the subjunctive? In Icelandic, subordinate clauses after verbs like say, claim, admit, think, etc. often use the subjunctive when reporting someone’s statement or viewpoint rather than stating it directly as plain fact from the narrator’s perspective.

So English usually just says had been, while Icelandic may use hefði verið.

Verið is the past participle of vera (to be).

It is used in compound verb forms, much like English been:

• hefur verið = has been
• hafði verið = had been
• hefði verið = roughly had been / would have been depending on context

So in this sentence, hefði verið works like had been.

Because þetta is not modifying lygi as an adjective. It is a pronoun meaning this, referring to the whole thing being discussed.

So the structure is:

• þetta = this
• hefði verið lygi = had been a lie

In other words, þetta refers to this thing / this statement / this matter, and then the sentence says that it had been a lie.

It does not need to agree with lygi here in the way an adjective would.

Because Icelandic has no indefinite article. There is no separate word corresponding to English a/an.

So:

• lygi can mean a lie
• lyg(in) would mean the lie if made definite

In this sentence, lygi is indefinite, so English translates it as a lie even though Icelandic does not use a separate word.

It is in the nominative, because it is a predicate noun after vera in hefði verið lygi.

So the idea is:

• þetta = subject-like element of the clause
• lygi = what þetta is identified as

This is similar to English this had been a lie.

Yes. A very natural alternative is:

• Þegar hann viðurkenndi að þetta hefði verið lygi varð samtalið vandræðalegt.

That means the same thing: When he admitted that this had been a lie, the conversation became awkward.

When the þegar-clause comes first, the main clause still follows Icelandic main-clause word order, so varð comes before samtalið in the main clause.