Do heroes get to keep the Blade of Light after the quest is finished?

Not to my knowledge no. You don't get to start Quest 7 with the Blade of Light.

You don't keep treasures from quest to quest, which aren't nearly as powerful as a relic...so why would you keep a relic? Also, considering I've played the later quests without relics and they still aren't that hard (certainly not compared to WoD, for example) it would way overpower the heroes.

You only start quests with your standard skills, and 300 gold for equipment, unless the quest specifies otherwise in the setup section of the quest (I think 8 lets you start with 500 for example and the ability to buy copper treasure). You start with the blade of ligth and you might as well just play with one hero.