You should get power back to the fuel pump when you crank the engine Tom and then for ~1-2s after. When you first switch the ignition on, the pump runs for 1-2s to prime the system hence the +12V on the pump feed.
While cranking/running, it gets a pulse from the CPS to tell it to keep going. Do the simple stuff first like charging the battery so you can crank the engine before overhtinking it!
Depending where the fuel hose you're replacing is, it varies in size, 12mm bore from the tank/pump outlet and under-floor pump inlet with semi-rigid nylon pipe in between.
The return line is 8mm and i can't remember if the hose between the under-floor pump and fuel rail is 8 0r 10mm bore. It should be J30R9 ethanol safe hose as you point out though.
Those sizes are for the standard car though, not sure if Neillsens altered any hose sizes when they built the hearse onto the standard car though.
As for the Lexus - bet you wish you'd thrown the car at a Lexus dealer now!
Hope you can get it sorted reasonably cheaply - windscreen clain on your insurance (£80 ish for the excess) and a used wiper motor with a few drops of ATF down the shaft before fitting and periodically after - new washer jet too! They have a similar system on Audis and even a 6 year old one had loads of rust on it!
