Car won't start
 

Brand new RX 350. Trying to start car get very rapid clicking but no starter turning. Emergency start procedure with ignition in accessory mode also fails to start. Battery is fine. Have Green light on Start button. Anyone have ideas?

I'd say about 90% of clicking but no start problem is battery related.
But if it is not the battery, you might wanna call the dealer.

Agree with Alexis-LS that its likely your battery. Even though the battery may seem good, its likely that one of the internal chambers within the battery is dead and hence not delivering the right amount of juice.

But then your RX350 is brand new! Was it sitting in the dealers lot for a while? - many months before you bought it? The registration sticker would tell you.

Went to car this morning and no power on at all. AAA came and jumped and car started fine. Let run for 45 mins and starts ok. However, now Check Engine light is on and Check AWD message is illuminated. Will be taking in to dealer tomorrow but the car was running for 90 mins yesterday before all this started so battery should have been fully charged. I suppose could be a defective cell.

Do you have another car? If so pull the battery in that car and switch it with the Lexus and see how it goes.