Excellent -- we've been thinking about this a lot on our end, and we'll be in touch via e-mail. Copy/paste between browser tabs/windows does work (uses HTML5 localStorage), but it sounds like you might be having a different issue and we'll follow up. Thanks!
Hey, another user here that would love to be able to just export parts of the circuit. I usually just use circuitlab for the rendering because SPICE is ugly and bloated and when in need of good looking schematics for reports circuitlab is great.
Maybe have a tool that allows you to select just a part of the schematic like some print screen tools have when you just want to save a part of the screen, and export just that. I never experienced the crashes that the other poster talks about though.