For the ultimate in insulation, get VIPs (Vacuum Insulation Panels). Very pricey, but R-25+ per inch is easy with them.
More realistically, use either rigid foam or foam in place foams. R-7.? per inch is the maximum I've seen for a spray in foam.
A reflective layer is a must. Either a space blanket or an insulation product like TechFoil. TechFoil is a multilayer bubble wrap with reflective layers in it. The main thing with reflective layers is you need to provide an air gap for them to work. A reflective layer with air on the hot side is generally found to be good for R-5 to R-7 against radiant heat.
On cooling the box. Look to marine equipment. Many 12 or 24VDC powered refrigeration and freezer units available, but expect to pay allot more than an ARB costs. ARB has volume of production. For efficiency, look at BTUs or Watts of heat pumped per Watt of electricity used for given interior and exterior temperatures. Also it is hard to beat ARB's efficiency.
As for powering it. No problem. For a few days, just put in a big battery bank that can be charged from the engine alternator or plugin charged at home. You might want to put in a higher output alternator on the engine, but... Think about this. If you are mainly a weekend warrior only going out for a max of 3 to 4 days. Something that draws 45AH/day* only needs 225AH of battery capacity for a full 96 hours (4 days) of run time. Remember deep cycle Lead Acid batteries should never be discharged more than 80% or you kill their life expectancy.
For solar you need a 3+ day battery bank for cloudy days. As for powering the Waeco CF-40. 45AH/day * 80% battery charge efficiency gives us 675Watt Hours solar to supply it for one worst case day. Figuring flat mounting the solar panel that means I'd need 200 to 250 Watts of solar panels for where I live to provide most of a worst case day. A 200 Watt BP solar panel is something like 62" x 32". Having two of them would be better as I could also run other stuff like my laptop and lights. For guesstimating how much you would get from a solar panel I refer you to solar energy sites. For short trips I don't think solar is worth it. Just use a big battery bank and alternator. For longer trips, yes solar is likely worth it. Especially if you wish to stay at one place for awhile. I've thought of mounting them to a popup tent roof of the style where one end is hinged and the other end lifts. You would then park facing south. Another thing is on light to moderate cloudy days the solar panels will output some power.
* Waeco CF-40 unit. 45 Watts @ 50% (a good worst case) is 3.75Amps @ 12VDC * 24Hours * 50% = 3.75 * 24 * .5 = 45Amp Hours @ 12VDC.
WAECO USA. With more normal conditions the unit should only use half to 2/3rds of this worst case.