> hann brá við öxinni

He parried with the axe; he warded off the blow with his axe, cf.
Zoega: bregða e-u við "to ward off with". If I've understood this
right, the weapon used to parry is dative just to express its role as
the instrument, rather than being governed by the preposition 'við'.
Instead, 'við' is adverbial, part of the phrasal verb 'bregða við'.
Does that make sense? 'hann brá við' "he reacted/responded, (or in
this instance, specifically) parried" -- the enemy's blow being the
implied complement of 'við' "against, towards, in response to" -- What
did he parry with? 'öxinni' "with the axe". I hope I haven't
ridiculously overcomplicated that...

> og kom í fetann og beit í ofan um tvo fingur.

"and [it, G.'s sax] landed in the axe-blade and cut down into [it] the
width of two fingers." Magnús Magnússon and Hermann Pálsson: "The
sword struck the axe-blade with such force that it bit an inch deep."

> Eigi ert þú engi í leikinum.

MM & HP paraphrase the idiom: "There's nothing half-hearted about your
way of doing things." (Lit. "You are not no one in the game.")

> "Far þú til Hrúts föðurbróður míns," segir hún, "og sjái (subj) hann

fyrir þér."

MM & HP: "... He will look after you." I think the subjunctive is
used here not so much to cast a doubt on the likelihood of his looking
after Th., but in the optative or imperative sense: "may he look after
you".