> Son hans hét Auðun faðir Ásgeirs, föður Auðunar, föður
> Egils er átti Úlfeiði dóttur Eyjólfs hins halta.

> His son was named Audun, father of Asgeir, father of
> Audun, father of Egil who was married to Ulfeid, daughter
> of Eyjolf the lame.

> His son was-called Auðunn, father of Ásgeirr, father of
> Auðunn, father of Egill who had (in marriage) Úlfeiða,
> daughter of Eyjólfr the Halting (Lame).

The nominative is <Úlfeiðr>. (The name is contracted from
<Úlfheiðr>, though I don't believe that the original form is
actually attested.)

> Annar son Ásgeirs hét Þorvaldur.
> Another son of Asgeir’s was named Thorvald.
> (The) second son of Ásgeirr was-called Þorvaldr.

I'd go with 'second', since a third is mentioned a little
later.

> Kálfur Ásgeirsson var þann tíma í förum og þótti hinn
> nýsti maður.

> Kalf Asgeir’s son was at that time on a trading journey
> and seemed the most ?? man.

> Kálfr Ásgeirr’s-son was at that time on trading-voyages
> (see för, Z1) and was-reckoned the freshest (newest?,
> superl of nýr?) person.

The past participle of <nýtast> 'to be of use' is <nýst>,
used here as an adjective; it appears to be pretty much
synonymous with <nytr> 'fit, usable, useful'. In the phrase
<nýtr maðr> Z. glosses <nýtr> 'able'; that would probably
serve here as well. It's definitely not a superlative.

> Hún var gift Þorkeli kugga syni Þórðar gellis.

> She was given (in marriage) to Thorkell kuggi son of Thord
> the loud.

> She was given-in-marriage (gifted) to Þorkell ‘Cog,’ son
> of Þorðr Roarer.

The usual word for a cog -- a kind of large merchant ship --
is the strong noun <kuggr>; the weak side-form <kuggi> is
actually closer to the Middle Low German <kogge> from which
the word was probably borrowed. I like 'Yeller' for
<gellir>: they're actually cognate.

> Ásgeir var mikill maður fyrir sér.

> Asgeir was a very accomplished man?

> Ásgeirr was a powerful man of himself (in his own right,
> see fyrir, Z.i.13).

I don't think that there's any 'in his own right' here: so
far as I can tell, it's the whole construct <mikill fyrir
sér> that translates to 'strong, powerful'.

> Það er sagt eitt sinn, að Kjartan Ólafsson byrjaði ferð
> sína suður til Borgarfjarðar.

> It is said once that Kjartan Olaf’s son began his journey
> south to Borg firths.

> That is said, one time, that Kjartan Ólafr’s-son
> set-out-on his journey south to Borgarfjörðr.(gen. sg)

Despite what the punctuation and English syntax suggest, I'm
pretty sure that <eitt sinn> refers to the journey, not to
the 'It is said', as if it followed the <að>.

> Þorsteinn kvaðst hyggja að hann væri góður drengur "er það
> vorkunn mikil frændi," segir Þorsteinn, "að þig fýsi að
> kanna annarra manna siðu.

> Thorstein said he thinks that he is a good brave man, “ it
> is, kinsman,” says Thorstein, “ that to you ( is) a wish
> to know customs of other people.

> Þorsteinn declared-of-himself to believe that he was a
> good (noble-minded) man “That is a great
> thing-to-be-excused, kinsman,” says Þorsteinn, “that (it)
> urges you (ie that you are eager) to search (explore,
> kanna not kunna) the customs (ways) of other men.

The sense of <várkunn mikil> (as distinct from the literal
translation) isn't entirely clear to me, but I suppose that
it's along the lines of 'thoroughly excusable', perhaps even
verging on 'requiring no excuse'.

> Kjartan kvað vel takast munu.
> Kjartan said ?? well.
> Kjartan declared (they?) will succeed well.

Plural <munu> is a bit odd, isn't it? I can only guess that
it refers to Kjartan and Bolli.

Brian