At the studio I worked at they were having a groupshoot once; one the attendee's AlienBee (B1600 iirc) set the diffuser on their PLM umbrella on fire. The plastic/fabric melted all around the modeling lamp and eventually caught the whole thing on fire. Hilarious stuff.
As far as using softboxes with hot lights, I know some manufacturers make heat-resistent (ie. expensive) softboxes. Off the top of my head I know Profoto makes a bunch, and Photoflex used to (not sure if they still do though). The photoflex softbox the owner had was used with a "Starlight" hot light (I think it was ~1000w) and never had any issues with the heat...looked it up once when she was selling some old stuff she never used and saw the softbox once retailed for $350 though. A bit rich for my blood, but then again I don't really shoot video. Browsed the profoto lineup recently and it's even more expensive. Whether any of that would work with your setup, I dunno....but I imagine it's at least a sign that there are options out there with constant lighting in mind.