"Do you think a thermal management is required at all?"
Yes, even if the CPU is not reaching its maximum operating temperature, it’s a good idea for significantly improving the longevity of the device. The TK1 will generate significant heat even at moderate load or at boot up.
"If yes, would the the heatsink as offered by Toradex be sufficient also in this environment (distance between heatsink and lid will be only one or two milli meters)?"
It is likely better than no heatsink as it will provide more thermally conductive material for heat to be dispersed over, but of course its ability to dissipate it any further is not great. Some will conduct into the air around the heatsink and a fraction of that heat will conduct into the case, but the temperature delta between SoC & case will be rather high. Providing any ventilation would at least provide some degree of convection - albeit there is only 1-2mm between heatsink & case. Or even better would be thermal coupling to the case.
"I suppose the best heat transfer could be achieved by somehow thermally connecting the TK1 to the enclosing metal box. This could be e.g. achieved with the Toradex heatsink and a gap pad."
You are correct. However, the Toradex Apalis heatsink would not be an ideal solution for this. An aluminum block with thermal pad on either side would better suffice. Ideally, such a block would also provide some form of mechanical mounting to the case or carrier board. However, if such a solution is not realizable, you would probably see some kind of improvement by placing a thermal conductive pad between the Toradex heatsink and the case with at least some thermal coupling for better thermal conduction as compared to air coupling alone.