You should be able to turn the engine off in any gear selector position, but you should not be able to turn the key all the way to the "lock" position (with the button just above the key). It shouldn't start in any gear posiion other than P and N.
There are no relays between the ignition switch and the starter solenoid, so the only thing that could be not working is the neutral start switch.
As for the key coming out, that is a common problem. I was able to get new keys cut at the dealer (they have to be done from the VIN, not copies of your old worn keys) to fix the problem on my Camry. The lock cylinder may also be worn enough that this might not fix the problem in which case you would have to get a new lock cylinder.
-Charlie
PS. Having driven manual trans cars pretty much all of my life, pulling the key out has stumped me on a few occasions (the shifter must be in N or P to remove the key when everything is working). Later I noticed the car was in D...