Mythopoeia

I somehow agree with Rune upon the matter of the Fire-giants and the Balrogs. The actions of the former during the Ragnarök when marching upon Bifröst and it breaks and falls simply remind me of the events in Moria, on the Bridge of Khazad-Dûm.
Adding another point to the matter of the relationship between the two entities: the first stronghold of Melkor, Utumno, resembles the realm of the Fire-giants, Muspellheim. With all its forges, subterranean fires and chambers of poisonous vapours, Utumno (along with Angband) manifests as the mythological and the philological counterpart of Muspellheim, which is 'Land of Flame' (not unlike a chasm of molten iron and eternal streams of flame), hence Utumno and Angband being 'Great Deep' and 'Hells of Iron'.

Moreover, there seem to be no restrictions to Surt being the inspiration for (I believe) both Melkor and Gothmog. As stated in The First Book of Lost Tales, Gothmog the Lord of the Balrogs, was originally described by Tolkien as being the son of Melkor. So the Valaraukar were intended to be the progeny of the Dark Lord. Furthermore, there is no denying that the Fire-giants may at the least bear the blood of Surt. And as Surt rules over Muspellheim, Melkor rules over his strongholds.
In Dagor Bragollach, the intensity of 'Fire' as a dominant element in the characteristics of Melkor is present, along with some points on the similarity between the Dark Lord and the Muspell-lord. The scorching of Ard-galen by the flame rivers may reflect the deeds of Surt as he sets the worlds, and Yggdrasil ablaze. Though the World Tree never succumbs to the Fire-giant's menacing actions (unlike Ard-galen), it is 'burning' nonetheless. And last but not least, the Dark Lord's colossal figure when confronting Fingolfin the High King at the twilight of the battle resembles Surt's being as a Giant - a Giant of Sable Fire.

I may have missed many more counterparts, but I will leave the matter for someone else to state them.