You are correct, the larger piece of metal inside the sensors should both be on the "inside" of the board, nearer to the middle and the W&L logo. If you move them to the back, the same applies. Sounds like you are good in this regard.
The second LED should be solidly on until you hit "Go" on the programmer webpage. Then, it should change state every time the clock square has 8 transitions (black->white or w->b). You have good insight that it sounds like a bad sensor on the data side. If there was a bad sensor on the clock side, then the second led probably wouldn't be flashing as it is. The sensors I have sitting in front of me are all clear, maybe one of yours got scuffed up in shipping? It shouldn't matter, I think.
I would hold the sensors so they are touching your monitor (or very close). That's always worked for me.
When the 3rd and 4th LEDs start flashing in the alternating pattern, that indicates a checksum error. The data to transmit is wrapped into "records" containing 16 bytes of actual data, along with the address where they should be placed, and a checksum. The checksum is a way for the Blinky kit to double-check that there weren't any errors during data transmission. If it detects a checksum error, it gives up and starts the blinking on leds 3/4.
I believe the blinky can operate down to 1.8v supply, so your rechargeable batteries should be fine. Thanks for trying them with fresh 1.5v cells too.
At this point, since you certainly sound confident in your soldering/desoldering abilities, I'm going to send you a few more sensors to try out. I'd try replacing the data sensor first, since the clock sensor seems ok. Please email us your mailing address to email@example.com
and I'll send those out today.
Thanks for your patience as we sort this out. It's incredibly useful to have a skilled solderer such as yourself help us troubleshoot wonky kits.