Your wiring to the remote start system is not correctly installed or your starter wouldn't ever try to work with the engine running. disable the system by pulling the fuses to it, or removing the power line-damage can occur with a starter trying to start while the car is running.

The wiring needs to be investigated, someone has connected the wrong wire, to the wrong thing or left out a relay that would protect your car.

Also, it sounds as if the auto programming, for the remote start is turned on to a preset time, causing the engine to continue running. Read the owners manual and turn it off.

If you didn't get the system installed and don't have the manual-scrap this system and have one properly installed. It's not worth the potential danger to your car. What else was left off? Was it securely and safely wired? Not worth the risk.

Good Luck. (I love remote start systems, but I solder every wire, test using nothing but digital voltmeters and led test probes to prevent damage to the cars electrical system-hire a pro.)

Kinda simple, don't know if you fixed it yet. your starter circuit is getting signal at all times. The reason you hear the noise in N or P is because that is the only time the nuetral safty switch is being engaged. I sure wish I could call you and explain it it woud be so much easier.

Try this, power from batter goes to ignition switch, when you turn switch it connects the power lead to the start output lead. That start output lead goes to a relay. a relay is a switch, it switches when it has power on one side of a coil and gnd on the other. When it switches connection is made from your supply of power fromthe ignition switch to the starter. Things that keep the relay from switch is what gear the tansmission is in. when the transmission is in P or N there is a switch usually connected to ground that supplies the ground for your starter relay. Hence put the car in to P or N and if the wire from the ignition is powered the starter will turn over. Point being have someone check your ignition switch. and if it isn't that, then the start circuit completely.