Bob, to check the bridge, just unhook it and use a multi-meter, set on the diode setting (I think they all have that feature nowadays) and check that one way each diode measures ~0.7V, open the other way. Do this for each diode. Very straight forward.
Just guessing, but I wonder if you connected the current probe across the output instead of series? That would be the same as shorting the output. Just a WAG. Got a picture of how the current meter was connected?
Was the fuse a slo blo or fast?
Regarding the 35V and the LM350K. The spec sheet only mentions the in-to-output voltage delta, not the delta to ground, so it might be OK. If you want to lower the input voltage, a simple way to drop it some is to add some series diodes. You can always use a spare bridge for that.