Would flicker shields absorb flux?
Well, I'd presume they'd follow the same rules in the CSV as the base shield, they'd just rotate and deploy a lot faster than usual, and their cooldown would mean less total Flux and greater dissipation of hard Flux under most circumstances, but leave the ship vulnerable when they're all in cooldown.
Think of them as a layered defense that can automatically provide protection, at the cost having a limited duration, but having the benefits and perils of automated threat response and better overall Flux management. Kind of like Accelerated Shields but with some additional penalties and benefits. They'd be pretty awesome vs. isolated strike or burst attacks but really lousy vs. sustained fire. So, vs. missiles or fighter flyby, they'd be really useful... against that Cruiser with a bunch of Tac Lasers... not so much.
If the ship had multiple shields, wouldn't they all just go down at once the outermost one overloaded?
Sure, they'd have to follow the rules for Overload. But, due to their limited duration (I'm envisioning them with 1 second or less) that's much less of a problem, because during their cooldowns they're not "up" and hard Flux can be lost.