Byltan var alvarleg.

Because -an is the suffixed definite article. Icelandic usually puts the on the end of the noun instead of using a separate word.

• bylta = a fall / a tumble
• byltan = the fall / the tumble

So byltan is the definite form.

Bylta is a feminine noun.

That matters because Icelandic adjectives often change form to match the noun’s:

• gender
• number
• case

Since byltan is feminine singular here, the adjective also appears in a feminine singular form: alvarleg.

Because the adjective has to agree with byltan.

Here the noun is:

• feminine
• singular
• nominative

So the adjective also takes the matching form:

• masculine singular: alvarlegur
• feminine singular: alvarleg
• neuter singular: alvarlegt

That is why the sentence uses alvarleg.

Var is the past tense of vera, which means to be.

In this sentence:

• er = is
• var = was

So:

• Byltan er alvarleg. = The fall is serious.
• Byltan var alvarleg. = The fall was serious.

Because Icelandic commonly uses a suffixed definite article. Instead of a separate word before the noun, the noun itself gets a definite ending.

So English:

• the fall

becomes Icelandic:

• byltan

This is one of the most noticeable differences from English.

It is in the nominative case.

That is because byltan is the subject of the sentence:

• Byltan = the subject
• var = was
• alvarleg = serious

In simple statements like this, the subject is usually nominative.

Because Icelandic predicative adjectives also agree with the noun they describe.

In English, adjectives do not change:

• the man is serious
• the woman is serious
• the problem is serious

But in Icelandic, the adjective changes form even after to be:

• Maðurinn var alvarlegur.
• Konan var alvarleg.
• Vandamálið var alvarlegt.

So even though alvarleg comes after var, it still matches byltan.

No. This is very normal Icelandic word order.

The sentence follows a basic pattern:

• Byltan = subject
• var = verb
• alvarleg = complement/adjective

So it works much like English:

• The fall was serious.

This is a straightforward subject–verb–complement sentence.

A rough English-style approximation is:

BIL-tan var AL-var-leg

A few notes:

• y in Icelandic does not sound exactly like English y; in modern Icelandic it is pronounced like i.
• Stress usually falls on the first syllable of each word.
• ll in alvarleg is just a normal l sound here; the unusual Icelandic ll pronunciation happens in some other words, not this one.

So the main stresses are:

• BYL-tan
• VAR
• AL-var-leg

Yes, but it would not mean exactly the same thing.

• Bylta var alvarleg. = A fall was serious or Fall was serious depending on context
• Byltan var alvarleg. = The fall was serious

So the version with -n is definite, while the version without it is indefinite.

No. Alvarleg(ur/t) is a general adjective meaning serious and can be used in many situations, for example:

• alvarlegt slys = a serious accident
• alvarleg veikindi = serious illness
• alvarleg staða = a serious situation
• alvarlegur maður = a serious man

In your sentence, it simply agrees with the noun and means serious in context.