Not long ago I did a small test to see how hot the base/web area of the cases got using a socket in the cordless drill and a torch. I also did the same test with brass straight out of my AMP annealer.
I tried 223, 22-250 and 257 Roberts brass with the torch and the bases attained a temperature of between 201°F (223) and 250°F for the 257 Roberts, the 22-250 was around 235°F. After the cases were annealed I placed them on a piece of wood with a Fluke temp probe between the wood and case.
When I did the same test with the AMP I only did 257 Roberts cases and the reached 230°F so pretty much in line with the torch and socket test.
This suggests to me that you can't hold onto the base of a case with bare fingers and anneal the case effectively.
As for cooling I drop my cases onto an aluminium tray which acts as a heat sink and stops the cases from scorching my wooden bench.