It's been 5 and a half months, but after another E3D visa holder got her work permit in only a month, we figured it was time to stop waiting patiently.
We went to the USCIS office in San Jose this morning to politely go "what the fuck?", and to my amazement, they didn't hesitate in issuing Sarah with an 8 month work permit on the spot. Apparently the full-blown one will come when its good and ready, but in the meantime, she's all good to go. Well, she can go try to get a Social Security number in 10 working days and see what happens...
While we were in the vicinity, we also went to the San Jose Social Security office, so I could have another go at applying for a Social Security card, after what happened last time.
Again, we waited for an hour, only to be told that no, they didn't keep my immigration verification on record, and I wasn't showing up appropriately on their computer system, so they'd need to re-request verification of my status from Immigration anyway. Argh!
Apparently this happens to other people, not just me, so it seems to be luck of the draw as to whether your vital statistics actually propagate through all the various Government systems or not.
So $DEITY only knows when or if I'll receive my Social Security card. At least I'm not sweating on it to get paid or anything.