Hi Rod,
Q1
The name "FM band" is misleading, since one can transmit FM on any frequency. All of these bands mentioned are in the VHF band which extends from 30 MHz to 300 MHz. In some countries FM broadcast radio is referred to as "VHF", "UKW" (German "Ultrakurzwelle" — "Ultra Short Wave") or "УКВ" (Russian "Ультра Короткие Волны").
Q2
AM ( LW/MW) is not very good so stick to FM the "aerial" is integrated into the rear window demister element and there is an amplifier mounted in the boot for this system.
Q3
The LS400 is so reliable that if serviced and driven correctly will never break down eliminating the need for an engine bay light.
Q4
Is that the one where you have to keep the button pressed down on the remote?
Q5
If you have a neighbour like that I would put the car straight into the garage so he cannot comment,you need to remember you have the Lexus he has the jealousy, seriously you will need a series 4 owner to help with the mystery box.
Q6
You can get battery chargers that connect with the battery still connected have a look in Halfrauds then buy it cheaper online.
Q7
You are talking about pulling the dash and centre console on a series 4 LS400 I wouldn't fancy doing that.
and finally,
After owning mine for over a year I spotted a black cylindrical plastic tube under the dash on the passenger side and with help from fellow members was told it was the "Flare Holder" mine is an import and a flare is a legal requirement in Japan should you break down.
Anyway got to go now I'm missing Coronation Street
PS post some pictures of your car up