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.
Good ideas.
Remember to have rational size fuses at the supply end of your power cables to protect against fire due to shorts in the walls. Use the thickest wire for the power pair that you can afford and draw through the walls with ease to minimise losses.
You can add a 48->5V converter for your RPi and perhaps a higher power 48->12V converter for your IR illumination if suitable. Such converters are commodity items.
You could use a second Cat6 cable for the power and use two pairs for ground and two pairs for +48V but it will still be less copper than a dedicated pair of the same overall size.
EDIT:
As clarified by Golanz in the comments above the voltage converters should be of the switching type and not simple linear regulators so the current in the 48V link is reduced, a linear regulator would dissipate lots of energy as heat in the regulator and cause totally unnecessary load on the power supply and wiring by the same amount. Finding linear regulators with a 48V input for 5V output would be harder these days anyway and only appropriate for rare analogue applications.
Best Answer
Yes you can deliver 50W over a cat 5 cable, the POE standard supports this:
Source: IEEE 802 3bt DTE Power via MDI over 4 pair task force
There are two ways to do this:
Source: Infocellular
So you would want to use the first method, with a step up DC/DC converter to boost the voltage to 48V. On the device end you can use your step down DC\DC's. The voltage is boosted to cut back on resistive losses in the cable. There are transformers that block DC on ethernet devices so your probably not going to get into too much trouble if you mix things up.
If you want it to be compatible with standards, then you will want to use the appropriate detection circuits. Usually POE uses rectifiers on the device end so either AC or DC can be used, but you also have a diode drop. If your running your own custom solution and pay attention to polarity then you would not need this.