Usually when something goes wrong after a job it's poor workmanship, unless the problem was there and you only just noticed it because you're paying more attention now after the work (this happens often too).

I'd say the tech messed up some wiring.

 

to me also it sounds like electrical issues after installation of new device ... either something was knocked loose or is shorting.

 

I'd assume the remote start system taps into the neutral safety switch (so the car won't start in gear)
I've never tried to set my cruise control in any gear but Drive, it may not work in lower gears. I bet it shuts off in neutral. Engine would blow if it didn't
OH!! I wonder if they just shorted the neutral safety switch? Try starting the car in drive (while you're sitting in the drivers seat with one foot over the brake pedal) If it starts, boy howdy did they mess up that install....

 

Yes, they are all related to the "remote start" installation.
All related to ECMs(computers) ability to detect status of transmission.
1. Neural switch code--transmission selector
2. no cruise control---transmission selector or brake issue
3. no "P" (Park) indicator--transmission selector

You will need you get the original installer involved, not just the company that did the install, you need the guy that did the work.
He needs to go over each step again and check to make sure he tapped into the correct wires.

If this is not possible, then you will need to get the instructions from the the installer including ALL manufacturer/model numbers of any parts used.

The added circuit board of the alarm/keyless entry/remote start does not work with all cars models the same way, there could be an "updated" installation procedure for your model they didn't know about.
So check with the maker of this system.

Most add-on systems are "piggy backed" to the cars existing circuits, this is fine unless their detection circuit interferes with the original circuit, which reads like what is happening.

But for sure the 3 known issues are related to the added system.
Only exception I see is the "D" light not working, that is probably just the bulb, it's the one that is used the most so the one that burns out first
But if you are just now noticing that then have them replace it as well