> og var sinn vetur hvort, utan lands eða með föður sínum.

"and he spent the winters alternately abroad and with his father" (one
winter abroad, the next with his father, and so on). Don't forget to
check the notes in Gordon!

> kom Bjarni skipi sínu á Eyrar
> Bjarni came in his ship

More literally: "Bjarni brought his ship to Eyrar". 'koma' is used
here as a transiotive verb meaning "to bring", the thing brought being

> Þau tíðindi þóttu Bjarna mikil
> These tidings Bjarni thought important

Yes, or maybe you could say that he was greatly affected by the news,
took it to heart.

> og þiggja að föður sínum veturvist

Lit. "and receive from his father hospitality-for-the-winter", i.e.
"spend the winter with his father".

> I am sure it is The Christ who is Blameless - innocent whereas I
first thought it applied to the Monks

Yes, you got it right in the end.