@vangstampede
Not sure if the police actually would. From inside, it looks like you can see the whole veranda through the glass doors anyway. If nothing strange appeared to be there, I can see why the police thought there's no point in looking more closely.
@assembly79
For the car to be running, the key would still need to be inserted into the ignition. If the key wasn't taken out, it couldn't have been used for anything involving the car door.
@Keimichi
It was very clearly stated Miyuki's currently working as a flight attendant, so at the very least, she's not dead. Plus, Hajime received texts from her at least twice during Uta Island Resort.