#1 possibly a defect in the "cold start" system - either the temperature
sensor is defective and always ON or the electric cold start valve is
stuck OPEN or,
engine's air filter is very dirty or air flow to filter or motor is
#2 I don't quite understand what you mean.
Motor is shut down, key is removed but 5 to 30 minutes later the
ignition is ON (by itself)?
If that's the problem it could be due to a worn out ignition switch or
defective (shorted) wiring harness.