Dagsektin byrjar eftir skiladaginn; skilakassinn er opinn.

Dagsekt is a fine or penalty charged per day. In a library or borrowing context, it is the overdue fee that starts accumulating day by day.

The ending -in is the definite article, so dagsektin means the daily fine or the overdue fine.

Even though English might sometimes use a plural idea like fines, Icelandic often uses the singular here to talk about the fee system itself: the daily fine starts.

Byrjar is the present tense of the verb að byrja, meaning to begin or to start.

Here it is:

• 3rd person singular
• present tense

So Dagsektin byrjar means the fine starts or the fine begins.

After the preposition eftir, Icelandic often uses the accusative when it means after in a time sense.

So:

• skiladagur = due date / return date
• skiladaginn = the due date in the accusative singular definite

That is why the sentence has eftir skiladaginn = after the due date.

This is a normal noun inflection pattern.

The dictionary form is skiladagur:

• nominative singular indefinite: skiladagur
• accusative singular indefinite: skiladag
• accusative singular definite: skiladaginn

So two things happen:

1. the nominative ending -ur disappears
2. the definite ending -inn is added

That gives skiladaginn.

They are compound nouns, which are very common in Icelandic.

• skiladagur = return day or due date
• skilakassi = return box, often a library book drop

The first part skila- is related to að skila, meaning to return, to hand in, or to submit.

So these are very transparent Icelandic compounds once you know the parts.

In Icelandic, the is usually attached to the end of the noun instead of being a separate word.

In this sentence:

• dagsektin = the daily fine
• skiladaginn = the due date in the accusative
• skilakassinn = the return box

This is especially important with compound nouns: the article goes at the end of the whole word, not in the middle.

Because opinn has to agree with skilakassinn, and kassi is a masculine noun.

So:

• masculine singular nominative: opinn
• neuter singular nominative: opið
• feminine singular nominative: opin

Since skilakassinn is masculine singular nominative, the correct form is opinn.

No, this sentence has very straightforward word order.

First clause:

• Dagsektin = subject
• byrjar = verb
• eftir skiladaginn = time phrase

Second clause:

• skilakassinn = subject
• er = verb
• opinn = adjective complement

So for an English speaker, the structure is fairly familiar.

The semicolon links two complete but closely related statements:

• Dagsektin byrjar eftir skiladaginn
• skilakassinn er opinn

The idea is: the fine starts after the due date; meanwhile, the return box is open.

A semicolon is a punctuation choice, not a special Icelandic grammar feature. You could also see these as two separate sentences in other contexts.