Cooked slices keep 3 days refrigerated and freeze 2 months with parchment between them. Reheat in a toaster or a 350°F oven; the microwave makes them limp. Don't store the custard with bread already in it.
Soaked too long, bread too fresh, or heat too high. Brief soak, stale bread, medium-low pan — those three fix it.
The dish long predates the name and exists across Europe as pain perdu, 'lost bread', because it rescued stale loaves. The English name's origin is disputed and probably not actually French.
Yes, up to a day refrigerated. Whisk it again before using, as it separates.