I can't understand the link between the drain cover screw and the water flooding? In order for a leak to happen pipes must surely have been split or knocked out of position.Did you poke anything down the plug hole to try to get the screw back,like a wire coathanger?
Surely if one knew the important role of the screw,you would not continue to let water pour down,and would insist on urgent repairs? I am assuming you only found out afterwards. If not,then I think your landlord could hold you responsible for paying at least the excess.
One doesn't usually have to mess about with mechanical fixings to eradicate a blockage. It's starting to sound like you did this to yourself... did you try any unblocker products first before you resorted to unscrewing things? What appears to have happened is that you undertook some (what I'd call) DIY and it went wrong - the loss of the screw - and then you just left it... I mean, that's harsh, because you told someone, but it's unlikely they truly understood what was going on (although someone would ideally inspect)... you continued as if nothing had happened... but you knew it had. You obviously knew that screw was there for a reason... mostly they [screws] are. Whether flooding is willful or accidental shouldn't really matter... because it was your actions that instigated it... unless you can prove otherwise, of course.
Oh well... at least you didn't come here with a fixed outcome in mind.