Nice idea,
Tuor! I always saw Túrin more as a blend of Kullervo and Sigurd the Volsung, but now you mention it, the parallels with Achilles look quite convincing, and I'd like to add another one:
8) Both show pity to an elder person they first perceived as an enemy - Achilles to Priamos, Túrin to Mîm the dwarf; in both cases after either the hero himself or one of his companions killed the elder guy's son. (Gosh, this sounds like One Thing in Common!
)
Interestingly, Achilles and Sigurd share the trait of invulnerability except for one weak spot, but I suppose Tolkien found this mythological motif too hackneyed and therefore decided not to use it.
Other classical parallels? Hmmm... If Tolkien had ever got round to telling the voyages of Eärendil before he reached Valinor, we might have a Middle Earth Odyssey. And the story of Beren and Lúthien could be seen as a version of Orpheus and Eurydice with roles reversed (note that Lúthien touched Mandos' heart by singing before him!) and Lúthien-Eurydice succeeding where Orpheus failed.