DOES THE TRUCK LID LIGHT UP OR IS IT JUST REFLECTORS ON THE 2013 DART RST

The trunk outer edge lights up when lights are turned on. Only the outside edge on the whole back around the backup camera lights up. Just turn your lights on and look at the back to see. :)

The optional light up rear deck is called the Race Track LED, it will turn on with the lights. Not all darts have this, it is an option package. but you can purchase it if you want it's around 280.00

Yes it depends on harness @ passengers reverse light if there's 3 wires ( racing) cool one 2 wires reflecting in sun light up

Why does my trunk button not open the trunk but every time I turn on my headlights the trunk opens up?

that's sounds like a wiring issue

rst 1.7 turbo? hate to tell you that dodge didnt make an rst 1.7 turbo... they made an sxt, areo and a rallye but they didnt make an SRT in 2013 and they never made a 1.7 turbo in any year. they made a 1.4, a 2.0 and a 2.5