As for

is sykomularnir just the icelandic spelling of sugarcubes?

It's ‘Sykurmolarnir’, but yes, it's a direct translation. Icelandic and English are comparatively closely related, so ‘sugar’ and ‘sykur’ are pretty obvious cognates, but in fact the word goes way back, and similar ones can be found in many other languages.

from Middle English sugre, borrowed from Old French çucre, borrowed from Old Italian zucchero, borrowed from Arabic سُكَّر (sukkar), borrowed from Middle Persian 𐭱𐭪𐭥 (šakar), borrowed from Gandhari 𐨭𐨐𐨪 (śakara), from Sanskrit शर्क॑रा (śárkarā), from Proto-Indo-Aryan *śárkaraH, from Proto-Indo-Iranian *ćárkaraH, from Proto-Indo-European *ḱorkeh₂ (“gravel”).

That's the Icelandic version and video for ‘Birthday’. The English video is rather more developed.

(I thought that the English video had a version with Icelandic vocals too, but can't find it on Youtube.)