Hann varð kvíðinn þegar hann sá hvað dagskráin var þröng, en hún sagði að það væri enn mögulegt.

Because verða means become, while vera means be.

So:

• hann varð kvíðinn = he became anxious
• hann var kvíðinn = he was anxious

The sentence is talking about a change of state, so varð is the natural choice.

Kvíðinn is the adjective meaning anxious here, and it agrees with hann.

Icelandic adjectives change form for:

• gender
• number
• case

Since hann is masculine singular, the adjective appears in the masculine singular form kvíðinn.

A useful thing to notice: the -inn here is part of the adjective form, not the definite article.

Yes. Sá is the past tense singular form of sjá (to see).

So:

• sjá = to see
• hann sér = he sees
• hann sá = he saw

This verb is irregular, so the past tense does not look very much like the infinitive.

It can mean both, depending on context.

In this sentence, hann sá hvað dagskráin var þröng can mean:

• he literally saw how tight the schedule was, or
• he realized/understood how tight the schedule was

That is very similar to English, where see can also mean understand.

Here hvað means something like how, introducing a degree/exclamative clause.

So:

• hvað dagskráin var þröng = how tight the schedule was

This is a common Icelandic pattern. A more explicit or formal alternative would be:

• hversu þröng dagskráin var

That may look easier to an English speaker, because hversu often matches English how more clearly.

Because dagskráin means the schedule / the agenda / the program.

• dagskrá = schedule, agenda, program
• dagskráin = the schedule

Icelandic usually adds the definite article to the end of the noun instead of using a separate word like English the.

So the ending -in here is the suffixed definite article.

Because þröng has to agree with dagskráin, which is feminine singular.

So the adjective takes the feminine singular form:

• masculine: þröngur
• feminine: þröng
• neuter: þröngt

Since dagskráin is feminine, þröng is the correct form.

That is just like English in many ways. Icelandic uses words meaning narrow/tight for time pressure or a packed schedule.

So dagskráin var þröng means the schedule was:

• tight
• cramped
• narrow in available time

It is a natural extension of the basic physical meaning.

Væri is the past subjunctive of vera.

After a verb like sagði (said), Icelandic often uses the subjunctive in reported speech, especially when the speaker is reporting someone else’s view rather than directly stating it as a fact.

So:

• hún sagði að það væri enn mögulegt = she said that it was still possible

If you used var, that would sound more like a plain factual statement. Væri adds a sense of reported speech, distance, or non-commitment from the narrator.

Because it agrees with það, which is neuter singular.

In this sentence, það works like English it in it was still possible. It does not refer to a concrete neuter noun; it is just the grammatical subject of the clause.

Since það is neuter singular, the adjective is also neuter singular:

• mögulegur = masculine
• möguleg = feminine
• mögulegt = neuter

Here enn means still.

So:

• það væri enn mögulegt = it would/still was possible in context, more naturally it was still possible

In other contexts, enn can also overlap with English yet, but here still is the best match.

Because those are subordinate clauses.

In a main clause, Icelandic often follows a verb-second pattern. For example:

• Hann varð kvíðinn

But after subordinating words like:

• þegar = when
• að = that
• hvað = how/what in this kind of clause

the clause usually has more ordinary subordinate-clause order, with the subject before the finite verb:

• þegar hann sá
• að það væri
• hvað dagskráin var þröng

So this sentence is a good example of the difference between main-clause and subordinate-clause word order in Icelandic.