I've had an LS430 which ended in disaster when the gearbox failed. Make sure you dip the transmission oil, it should be red and also make sure it doesn't smell burnt. I'd change the radiator as a precaution because the transmission fluid is also cooled by the radiator and if the radiator gets any corrosion it can allow engine coolant into the transmission fluid which can cause the gearbox to completely fail without warning. I think that's what happened to mine!!! Gear changes should be barely noticeable.
The suspension is prone to problems mainly due to water ingress into the ride height sensors and is VERY expensive to fix, some owners replace the air suspension with standard coil over suspension because it's much cheaper than repairing the air suspension. The ride should be smooth with no creaks or knocks, put the switch to high, the car should raise by about an inch and the compressor should be quiet.
Make sure both steering column motors work, again very expensive to fix.
Subwoofers can be repaired with varying degrees of success, new ones cost £350 or more from Lexus. Door speakers can blow too so take a CD with you with lots of bass to check all speakers.
Parking sensors are a common failure point.
The double glazed side windows can become delaminated with age.
Bonnet and boot support struts can be weak but are cheap and easy to replace.
The heated and cooled seats often don't work very well.
Check for correct red Toyota coolant.
Full service history is vital and check for cam belt changes every 100k or 10 years.
Check all electrics work and there's lots to check!!!
Check AC works in both front and rear if premium pack fitted.
Check no warning lights are lit on dash.
Check sat nav as this can go wrong.
Make sure toolbox is present and the locking wheelnut.
All the other checks you would do on any used car.
Personally after my experience with the LS430 I would be very nervous getting another, if I was going to get another Lexus I would find a nice mk4 LS400. I have no experience with LPG so can't comment on that. Lexus are great well engineered cars that do last well but they CAN go wrong and DO go wrong and can be horrendous money pits if you get a bad un. Tread carefully and you should be OK. Get it right and you have a bargain luxury saloon but get it wrong and it can be a whole world of hurt!!! Good luck!!!