Can I give you a precise burn number? No, but I sure as heck know the average fuel flow and I look at the enroute time and figure if the number looks right to me. I know my cruise speed, I know the distances. I do take the distances off the release, but I eyeball them and have a rough idea if they are close (I caught a major error in a trip in east Asia once). I don't need preferred routing, but I do look at the routing. I know where to find that stuff if I need it, and have done all of that on charters (including my own weight and balance by hand!). I look at the winds aloft and look at the computers wind component and see if they jibe.
Do I do it all to the exact detail? No, but I do know how to do it and would feel confident doing it (would take more than an hour prior to dept to get airborne, though!).
Finally, no, I don't study the AIM before each PC, but I do read it and the applicable sections of PANS-OPS once/year and also the company FOM cover to cover once/year. Over the years I have reached the point that I can skim many parts, concentrating on the changes. I also read through the intro section of the jepps.