you installer might not have hook up the factory alarm dis-arm wire up correctly, or the system does not have the option to pulse the disarm wire when opening the trunk.
this wire in the car serves three purposes,
1, disarms the factory system
2, unlocks the doors
3, rolls down the windows
so from the R/S systems brain you need to have three different wires connect to it and isolated with three diodes,
the first wire is your disarm pulse, if they did not do this right it should set the OEM alarm off when you Remote start the car,
the second wire is the unlock wire that should be programmed to two pulses so that it unlocks all of the doors the first time you press the unlock button. the third wire should be for an optional channel on the R/S system (optional for window control) and should be programmed to around 18-20 seconds and will simulate holding the key turned in the drivers door cylinder and will roll all of you windows down
the same should be done for the R/S systems lock, factory alarm arm, and another Aux channel to roll all of the windows up (something the stupid factory system cant do)
the wires in the car are the wires that go from the key cylinder in the drivers door (lock and unlock detection) to the Drivers Door Module.
now if it was hooked up but it still goes off, either it wasn't programmed to pulse right or it doesn't support that option and hitting un-lock on the system will also disable the factory alarm before you press to open the trunk.
5 days is totally unacceptable, especially for an experienced installer, My first LS took around 13 hours and that was figuring everything out like you need an extra relay for the ignition wiring, fuel door pop, the door trigger wire, and figuring out how to incorporate the factory window control. the second was done in around 8 and there was still differences to figure out because of it being a different generation ,while the LS is a little more complicated than most other cars, it is still a ford and pretty much the same as all new ford systems like the fusion, edge, taurus, F-series... the work is exactly the same with the exception that some of the wires are inside the drivers door and at the trunk modules instead of all at the SJB in other cars.