> "Eigi skal það," segir Þórður, "gakk þú til einvígis við mig."
> "Shan't " says Þorð "come you each one to fight with me (single combat)

I think 'skal' is impersonal here: "that shall not be", "that's not
going to happen", rather than "I shan't", although they add up to the
same thing as far as the ultimate meaning goes. Compare the formula:
'svá skal vera' "so shall/must it be."

einvígi, neuter sg. "single combat"
So, lit. "go thou to single-combat with/against me"

> lagði í gegnum hann.
> ran him through.

Exactly.

> Rannveig mælti, móðir Gunnars: "Það er mælt Sigmundur að skamma

stund verður hönd höggi fegin enda mun hér svo.

> Rannveig Gunnar's Mother said "That is said (proverb?) Sigmund,'

soon the hand that strikes is sorry' -and this will be so here""

Yes, proverbial: "for [only] a short time is the hand happy at the
blow", or as paraphrased here: "the hand's joy in the blow is brief" [
http://www.usask.ca/english/icelanders/proverbs_BNS.html ].

skamma stund "for a short time", feminine accusative; accusative used
for duration of time.
höggi, neuter dative sg.
fegin "glad, happy", feminine nominative (cognate with English 'fain').

> Well - it looks like here we go again - watch this space - shan't we

Indeed, although from the sounds of it, the feud is about to step up a
notch: this time it's personal...