Poly mouldings cut nicely with a saw blade; the fewer teeth, the better.
A chopper works, and makes cleaner miters, but I suggest making several small 'bites'. If you cut through too much at a time, some of the more-brittle plastics might break.
A manually-cranked sander will easily touch up imperfect miters, but a full-speed electric sander will quickly melt the plastic.
Join on the v-nailer, using cyano-acrylate (super glue). Note that accidental glue on the moulding's finish will ruin it. Ordinary frame glues, made for porous surfaces, will not work on polys.
Due to the density of polystyrene, v-nailing will often cause a gap on the outside of the miter. (MDF mouldings have this same characteristic, to some degree.) This may be avoided by gluing/vicing the corners before v-nailing. Or, if you work with only poly mouldings, slightly changing the miter angle will compensate, but then the angle would be wrong for wood & metal mouldings.
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"What would you do if you weren't afraid?"
I'd jump out of an airplane without a parachute. Sometimes fear is a good thing.