Here’s my translation

Alan

Búi bað hann hafa þökk fyrir og gaf Rauð fingurgull gott og mikið.
Búi bade him to have thanks (þökk, Z2) for (it) and gave Rauð a gold-finger (ring), good and large (the man with the Midas touch).

Rauður þakkaði honum gjöfina "og kom hér Búi," sagði hann, "ef svo ólíklega er að þú komir aftur."
Rauðr thanked him for the-gift „and come here, Búi,“ said he, „if (it) is so unlikely (ie in the unlikely event) that you come back.“

Búi kvað svo vera skyldu.
Búi declared (it) should be so.

Hann háttar nú svo öllu sem Rauður hafði mælt, kom jólaaftan undir þessa
He so arranges everything now as Rauðr had spoken, came on Yule-eve (accusative) under this

gnípu og dvaldist þar um hríð og sá þar ekki líklegt til dyra.
peak and dwelt there for a while and saw there nothing likely for (ie to be) a door-way.

Búi drap þá hjöltum sínum á hamrinum og mælti: "Þú Dofri," sagði hann, "lúk
Búi struch then with his sword-hilt-knobs on the-hammer (crag) and spoke: “You, Dofri,” said he, “open

upp þú höll þína og lát inn farmóðan mann og langt að kominn. Þar byrjar þinni tign."
upp (you) your hall (as in the hall of the mountain king) and let in a travel-weary person (man) and come (arrived, past participle)  from a long-way (langr, Z2) to (you). There (it) behoves (byrja, impers) your honour.”

En er Búi hafði þetta mælt þrem sinnum þá lét í hamrinum sem er gengur reið
But when Búi had spoken this three times, then (it) resounded (láta, Z11) in the-hammer (crag, precipice) as when a clap-of-thunder (reið, Z3) goes (claps)

og í því spratt í sundur hamarinn og urðu á dyr og því næst gekk kona í dyrnar.
and in that (instant? way?) the-hammer (crag, precipice) burst a-sunder and a doorway materialised (lit: came on, verða á, Z8) and in that next (instant) a woman walked into the-doorway.

Hún var mikil á allan vöxt. Hún var fögur að áliti og vel búin, í
She was large in all (aspects of) growth. (whatever way you looked at her she was a big, well-developed woman). She was fair of countenance and well-dressed, in

rauðum kyrtli og allur hlöðum búinn, og digurt silfurbelti um sig. Hún hafði
a red tunic and all (things, neut pl) adorned with lace-work, and a thick silver-belt about her. She had

slegið hár sem meyja siður er. Var það mikið og fagurt. Hún hafði fagra hönd
hair cut (pp of slá, Z4) as (the) custom (siðr, Z1, not síðarr) of girls (maidens) is. That was abundant and fair. She had a fair hand (fem acc sg)

og mörg gull á og sterklegan handlegg og öll var hún listuleg að sjá.
and much gold on (it) and a strong-looking fore-arm (masc acc sg) and all-(round) she was handsome to observe.

Hún heilsaði hinum komna. Hann tók því vel. Hún spurði hann að nafni.
She greeted the (not honum) arriving-one. He accepted that well. She asked him of (his) name.

Hann sagði henni "eða hvert er þitt nafn eða kyn?"
He told her “but what is your name or kin (I don’t think Fríðr´s gender, kyn, Z3, is in question here – they’re not in New York or Bangkok)”

Hún segir: "Eg heiti Fríður dóttir Dofra konungs eða því berð þú maður um herbergi vor?" sagði hún.
She says: “I am-called Fríðr (Beautiful), daughter of King Dofri, but for what concerning our lodgings do you, a human, endure (bera, Z6?)?” she said (what is it about our abode that drew you hither?).

"Eg vil hitta föður þinn og biðja hann jólavistar. Hann er einn frægastur konungur."
“I want to meet your father and ask (biðja not bjóða) him for Yule-lodgings. He is the most-famous king.”

Fríður mælti: "Ekki ertu ólíklegur maður að sjá. Kalla eg ráð að þú gangir inn með mér."
Fríðr spoke: “You are not an unlikely person (man) to see ( I´m not surprised to see you?) I call agreement (I affirm my consent, say it´s OK, ráð, Z5?) that you should-walk inside with me.”

Búi gerði svo. Fríður bað þá aftur lúkast hellinn og svo var. Þau gengu þá
Búi did so. Fríðr bade then the-cave to be-closed shut (lúkast, Z6 but cf lúka aptur, Z5) and (it) was so. They walked then

um stund og lýsti af eldi. Hún sneri þá út að bjarginu í einum stað.
for a time and (it) shone from a fire (ie a fire illuminated the way. I don´t think they were actively doing this). She turned then out towards the-rock (precipice = hamarr?, Z1 or Z2?) in one spot.

Var þar fyrir hurð og önnur og þá komu þau í lítið herbergi.
A door was there before (them) and a second and then they came into a little room.

Það var allt tjaldað og ágæta vel um búið.
That was completely hung-with-tapestries and excellently well arranged.

Fríður mælti: "Hér skaltu Búi setjast niður og hvíla þig og leggja af þér
Fríðr spoke. “Here shall-you set-yourself down and rest yourself (take a rest) and remove (lit: put off from you)

vopn þín og vosklæði."
your weapons (neut acc pl) and bad-weather-clothes.”