nanohertz, as to your questions:
1. About losing keys. Looks like x1le already answered your question, but if I can explain further and add some comments:
--It's less expensive if they use a universal bypass box, and one of your spare keys goes in there and stays there for the life of the installation.
--For extra money, they make modules that tie into the car's data wires, and emulate the "chip" that's inside your key. You pay more, but you keep all your keys. (Generally, these modules cost LESS than making more keys at the Infiniti dealer.)
--The fact that your installer didn't even try to offer you one of the no-key-required modules is a sign that he's allergic to money, or doesn't know what products are available for your car.
2. Your factory alarm.
If installed properly, the Avital (or any other brand) will arm your Infiniti alarm when you press the Avital "lock" button, and also disarm the Infiniti alarm whenever you press "unlock" or "remote start."
Your FX already has a good basic factory alarm that will honk the horn if a door or hood is opened, and the chip key already makes it hard to steal.
Adding an aftarmarket alarm gives you a shock sensor, as well as the options of things like a 2-way paging remote (like the 5303 has), and you can add various other sensors such as tilt, glass break, et cetera.
You could choose just a basic remote starter, and rely on your Infiniti alarm if you think it's good enough, or go all out and get something like that 5303 you had in mind.
3. Remote starter on friends' cars doesn't always work, especially when cold.
A non-defective, properly installed remote starter should be capable of starting the car under pretty much any conditions where you could start it by key.
Obviously, if the car is non-functional (dead battery, broken fuel pump, et cetera) the car certainly won't start, by remote or by key.
Hopefully your friends went to a reputable installer who warranties both his work and his product. They should arrange to bring their cars back and have the issue corrected.
4. Damages the car. You mentioned the starter motor and the alternator.
A properly installed remote starter probably engages and releases the car's starter motor even better and more reliably than you can do by hand.
Almost no remote starter connects to or gets involved in any way with the alternator. I see no basis whatsoever in this claim.
The only argument that could be made is that idling an engine, especially a cold engine, puts additional wear on it. Well, that's true of course....almost anything in life is using up some of its useful life while it's "on," whether it's a TV, a toaster, or a car.
And almost everyone these days that it's best for your car to get it warmed up as quickly as possible, which is accomplished by starting it, and as soon as it's running smoothly, driving at moderate speeds until it's warmed up.
Idling a car for fifteen minutes in the driveway every morning clearly is a slower way to warm it up, and in my opinion (not everyone agrees), surely puts a little more wear on the engine.
But, how long do you plan on keeping this car? Are you really going to keep it until it's so old the engine wears out?
My guess is your car will be in the junkyard someday due to an accident, or rust, or a handful of things wrong with it that just aren't worth fixing anymore, long before the actual engine wears out.
P.S. Does your FX have the Intelligent Key option?