I can't get there from my current location. Go to the12volt.com and find the section on relays. Look for a 5 pin relay. Typically a relay is used as a switch. The way a relay works is there is an actuator built into it, aka the switch (pins 85 and 86). When that actuator gets power and ground, it clicks and supplys whatever input (pin 30) you've give the relay to the 87 pin. Without power and ground both, the relay will let whatever your input (pin 30) is to flow through to pin 87a.
This means that if you:
Take the interior light power wire and put it to pin 30 (input) of the relay, this is your input or supply for the puddle and projector.
Take your door trigger and connect to pin 86 (pins 85 and 86 one needs to be power and one needs to be ground, doesn't matter which is which, 12volt might have them backwards to what I list)
If the door trigger is a ground, then give Pin 85 power. If it's power then give it a ground. You must supply the opposite of pin 86.
Now for the outputs. Pin 30 will automatically flow through the relay to pin 87a without the relay being actuated by power and ground. Connect 87A to your puddle light.
Now connect 87 to your projector. When the relay is actuated by the door trigger, pin 30 will be connected to 87.
This sounds way more confusing than it is..which is why I suggest a minute of reading on 12volt.
