When dropping with death rain, he makes a free barrel roll, the question is"Can he barrel roll before the bombs detonateand quickdraw shoots or does it have to be after?"

Short answer yes.

if it's a detonate at end of activation you'd move befor the shield was pulled.

If it's a detonate on overlap you get to choose what order the good things happen so drop barrel roll then detonate.

Deathrain (PS6) - When dropping a bomb , you may use the front guides of your ship. After dropping a bomb, you may perform a free barrel roll action.

Quickdraw (PS9)- Once per round, when you lose a shield token, you may perform a primary attack.

Assuming a regular bomb with Deathrain, you reveal your dial, drop, the bomb, barrel roll, and perform the movement. Regular bomb goes off after all movement is complete.

Assuming an "Action" bomb with advanced sensors, you drop the bomb - at that point it triggers the barrel roll and an overlap that would set off the bomb, causing damage at the same time. Since these happen simultaneously, my assumption is that the player dropping the bomb chooses the order they resolve. - So I agree with GrogEgrog.