We've just posted Adobe Illustrator (.ai) versions of the 2V and 3V dome files, along with an assembly instruction sheet we designed for a recent event (PDF). Note that the panels are scaled for 24" x 36" cardboard sheets--you may need to scale them to fit your laser cutter, or your printer if you're planning to cut the panels by hand.
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That'd be a fun experiment! Given all the holes and seams, I bet the dome leaks like a sieve.
You laser-cutter can absolutely cut the cardboard and foamcore board. The trick will be scaling the panels to fit in your machine. We've posted the CorelDraw files. Just make sure you proportionally scale the part (ie x and y axis scaled by the same amount) so that the geometry still works.For extra credit, you could use the Excel spreadsheet to calculate custom panel dimensions that are a better exact fit for your machine.