Yes, you really want to ground the strings, so a ground wire out to the tailpiece is required. It really is easiest to push this through and then fix it from from the outside rather than try and find a small hole in the rear block and push it through from the inside via an f-hole. So you may find it easier to either remove the existing wire for this purpose from your harness, then use another piece to push through which you should be then able to pull through the f-hole and solder onto the rear of a pot.

(Which reminds me that the back of the volume pot doesn't appear to be connected to ground, so the casing won't provide any shielding for the innards).

The easiest option might be to keep the harness as it is, push another bit of wire through for the tailpiece grounding, pull the ends of both wires through the f-hole and then solder the two wires together (cutting down the length as necessary first so you don't have an excess of wire inside the guitar) after first slipping a piece of heat shrink over one end so that the job can be insulated to avoid any accidental shorting.

I'd be tempted to cut off the unused connection tab on the tone pot. It's not ever going to be used but it might get bent when you are fitting it into the guitar and short out on the casing, which would stop things working properly. It's almost impossible to see inside the body to check that nothing like that has happened.

For the tailpiece connection, I'd suggest drilling a hole that's just big enough to get the wire through, then drill a short bigger hole (like for a rear ferrule on a Tele), so that you can tie a knot in the wire that fits in the hole and stops the wire ever being pulled through. The end then either needs to be stripped and flattened so that it sits pressed against the underside of the tailpiece, or soldered to a small washer that is held in place underneath the tailpiece by a fixing screw, or even just has a loop soldered in the wire which fits around a screw.

You can always test your wiring harness out by holding the pickup above another guitar and strumming it, after connecting the jack socket to an amp with a lead. A second person comes in handy for holding the pickup whilst you strum and turn the pots to check operation.