few 12 volt SLA batteries
You need two batteries. One for positive supply, one for negative. For positive, connect the "+" to VCC and "-" to GND. Negative, connect "+" to GND and "-" to VEE.
chassis ground connected to the RCA's ground for input/output
Chassis should connect to earth ground. You can float the rest of the audio circuitry or connect it to chassis (depends on rest of audio system). I would try connecting it first. This is probably best done at the S/PDIF input, isolate the audio outputs. Once the RCA shield is mounted to chassis, the rest of HAGDAC board is referenced to this point.
little boxes on the wiring diagram
Those are just no connects.
pins 19 and 20 to be connected together
Yes, this implements fast rolloff. Leave unconnected for slow rolloff. Many people wanted this option. We found the detail and sonics to be better in fast mode, which really suprised me. I expected otherwise.
attach LED's to the RLK and PLK pins
Yes. Look at the CHIME schematic for ideas. It is not the only way to do it. These are open collector logic levels, so they are pretty easy to interface with. The simplest being an LED from pin (say PLK) to ground. The pullup will drive LED - but you'll need more current, so add another pullup in parallel.
One note of caution, with PLK, it also is used to MUTE the audio for non-44.1k inputs. The LED voltage might be too low to signal a high logic level. Without muting, at 48k word rate, the output screeches loudly. Using a blue LED would be ok. Here's one way to do it with 12V batteries

How would I implement a polarity switch?
It is just a switch to ground. See CHIME schematic.
Metal cases are generally better for audio, but in this case I think you can get away with plastic.
jh
