When a song card is attached to a hero, that hero gets specific song sphere only for paying rsources or it also get that sphere trait ? For example: can we attach A Burning Brand to a hero that does not belong to Lore sphere, but has Song of Wisdom attache to it?

Yes, when a hero gains another sphere through a card effect it is considered to be a hero of that sphere and its original one. So Burning Brand/Song of Wisdom would work.

Some cards reference having a "printed" sphere, though. Then the hero has to be native to that sphere, card effects won't help you.

And also, bizarrely, some cards are also "considered to have the printed <X> icon" (Gandalf, spirit Bard) based on their card text and not their printed spheres. Bizzare. All they did was create a tier system for sphere icons.

Thanks!