This sounds like a really odd combination of symptoms, but they do share one common component - The Intelligent Power Module, which is located under the hood, on the driver's side, next to the relay box. If you have an intermittent problem, then maybe you have a loose connection here.
Another potential problem area is the ignition switch itself. There are at least 3 main circuits that feed through the switch and if there is a bad contact inside the switch, it is possible to have one or more circuits not working when the switch is in the "run" position.
Hope this helps,