You can use the mana produced by Brotherhood Headquarters’s last ability to cast a spell with freerunning even if you’re not paying that spell’s freerunning cost.

“Assassin sources” include any objects with the Assassin creature type. This means you could spend the mana to activate an ability of a permanent with changeling (a keyword ability that gives it all creature types) or an Assassin card in your hand or graveyard, for example.