Heill Dan!

You did exceedingly well as usual.

> kømr: strong verb, 3rd person, singular, present tense,
> indicative, active

Correct.

> Norðrlönd: strong noun, neuter, plural, accusative

Correct.

> tíðenda: strong noun, neuter, plural, genitive

Correct.

> Ólöfu: strong noun (proper name), feminine, singular, dative

Accusative - see next letter.

> dóttur: unique noun, feminine, singular, accusative

Correct.

> Þorsteins: strong noun (proper name), masculine, singular, genitive

Correct.

> rauðs: adjective, masculine, singular, genitive

Correct.

> þeim: pronoun, masculine, singular, dative

Plural, as you spotted yourself. It's interesting that
Arlie made the same mistake. I guess you were both thinking
of the dative masculine singular of the pronoun 'sá'.

> vildi: irregular verb, 3rd person, plural, past tense,
> subjunctive, active

Correct.

> kaupa: strong verb, infinitive

Correct.

> "Vildi kaupa" is in the subjunctive mood, which means that
> it is indicative of a hypothetical outcome. Hólmgeirr doesn't
> know what they want to buy, whereas, in sentence 1, the
> brothers have a definite wish, thus it is in the indicative
> mood.

Yes. A more specific rule is that clauses introduced with
'spyrja' have their main verb in the subjunctive mood.
Some real examples:

Hann spyrr, hvat þeim væri þar gefit.
Jarl spyrr Gunnlaug, hverr hann væri.
Hann spyrr, hvert Sigurðr skyldi fara.

But I guess there are exceptions.

Hrómundr spyrr, hverr nú vill ganga í hauginn

> A man is named Eyvindr - he lived in Norway.
> Maðr er nefndr Eyvindr - hann byggði í Noregi.

Very good! (In my new spelling standard the 'o'
in Nóregr is long - but Noregr is a perfectly
valid form.)

> His brother was called Gunnarr.
> Gunnarr hét bróðir hans.

Very good!

> The brothers were both big and strong.
> Þeir brœðr váru bæði mikill ok sterkr.

The adjectives should have the plural form
- other than that these are exactly the words
I was "fishing" for.

You seem to be using Microsoft's oe-ligature character
(not part of the latin-1 standard). That's something I
don't recommend - as it is difficult to view for users
of non-MS systems. Another reason for avoiding it is that
in italics it looks almost exactly the same as the
italic ae-ligature.

Kveðja,
Haukur