Ford Explorer R/S

My guess would be that the "factory alarm" isn't the problem. Since when you are remote staring the car the alarm isn't going off (horn honking), It most likely doesn't  have a factory alarm.  In order to check if it has a factory alarm do the following:

1) Roll down the window

2) Close doors and exit car

3) Lock the doors using the  Factory keyless entry remote.

4) Wait about a minute or more, then reach in the car through the window and unlock the car and open the door.

5) if the horn doesn't honk then you don't have an alarm, If it does honk you need to hook up the factory disarm wire of the remote starter to the  DARK GREEN/ PURPLE (-)  in the DRIVERS KICK PANEL

But like I said that isn't going to be your problem. The reason it would start only with remote start and not with the key is that your bypass (omega diagram) is coming on at the wrong time.  Its only supposed to be activated only with remote start.  If it activates also when using the key, the car will "see TWO keys" and not start.  Since when you are remote starting the car the 2nd key is not near the key cylinder it starts.  When you try to start it with the key, the car is seeing the key in your hand and the one hiding under the dash and won't start.

When you hooked up the relay according to the diagram make sure of your connections at the relay, they should be as follows:

Pin 86  - "Ignition 3" output or "Status output" or "negative while running" from the remote starter

Pin 85 - Constant +12V

Pin 30 - One side of the wiring that is around the key

Pin 87 - The other side of the Wiring around the key

Pin 87a - the middle pin is not used

After you verify the wiring,  just make sure where you hide the key under the dash is not near the ignition cylinder! 

Good Luck, Frank