Passive POE: Need 5V @ 2A at the destination
Let's start with a couple of straight answers: Good choice, possible, maybe, correct, yes, unless they use AWG26 or 28, which some flexible ones do, that's not how that works.

I'm betting you want more, right?

So, starting with your "biggest mistake": Power output at the end is not what a cable cares about. Cables care about current. Current creates a voltage drop caused by the wire-resistance. This voltage drop multiplied with the current creates power loss in the cable, which creates heating. Heating is bad. Bad is not good.

If you want to stick to "safe practise" you need to stay at the current level described in the standards, not the power level. So to transport your 10W, you'd need: V = P/I = 10/0.36 =~ 28V

But, that doesn't account for power loss in a conversion. If you use a buck regulator, which actively turns a given voltage into a stable 5V very close to the device, it can give you an efficiency of 85% or more, depending on which you choose. So then you are allowed to estimate using: Pin = Pout / efficiency = 10W / 0.85 =~ 11.8W.

Then you need to put V = Pin / I =~ 11.8 / 0.36 =~ 32.7V into the converter.

Because you still have wire losses you will need to put in a little more, so round up. Let's say 35V to be safe. Because of possible spikes and low-power-use moments your buck converter will need to be able to convert 28V to 38V without problem if your 35V is a clean voltage. Direct car voltages are unclean and very dangerous to shop for!

You can then create the boosted voltage from your car voltage with a boost converter that turns 9V to 34V into a nice stable 35V. Make sure it's rated for car voltages, or search for questions about "how to protect my device from car voltage spikes and load dumps".

Of course you can also just use the 48V that most early PoE systems use and get a ready made PoE to 5V converter on your device, then you just need a booster for car voltage to stable 48V.

But be very carefull in wiring all of this, make sure you look up how to transport a DC voltage over network cables, if you do it wrong your transmitter and/or receiver magnetics may melt.