Related problems? (Locked steering wheel / Non-start)

[Antharro]

My LS430 has two problems and I'm wondering if they're related.

1) Sometimes, the ignition switch won't turn, regardless of if the key is in it or not. The car definitely recognizes that the key is present. I understand this is related to a sticky "Column Tube Stopper", which I assume should unlock when the car either detects the key either physically or electrically present. A hard jerk of the wheel in each direction seems to fix it. I would like to find a permanent cure.

2) Occasionally, the car won't start. It will turn over and turn over and turn over but won't fire. Priming the fuel pump a couple of times makes no difference. When it eventually fires up, it's on all eight cylinders, no skipping, mis-firing or other problems, smooth as it should be. It only does this when hot.

Is it possible that these are both electrical problems relating to the car not fully de-immobilising? Or something about the key detection not working and therefore not letting the car run?