Well unless your apt. has specific rules on what kind of mini blinds, I'd just get a stock pile of the $5 ones from Lowes and just replace the ones that get ruined.

I have the same problem here at home, from the sliding door verticals being fine, to cords tied up and sticks removed. I feel your pain there.

I never have figured out a way of keeping the cats out of the blinds. My RB cat Dusty rarely ever bothered then, she'd just nudge them out of the way and go behind them, all the rest either play with them or plow right through them. *sigh*