Questions about Intel Core i5-6600T

No, it even comes without a box, packed in ESD foil / bubble wrap.

The cooler does not actually have to press so hard on the processor. The most important thing is an even layer of heat-conducting paste. I would screw in the screws so that the pressure on the cooler is exactly between "lightly applied" and "board bends". If the fastening screws have springs (springs press on the cooler holder), then the screws do not have to be fully screwed in, the pressure of the springs is already sufficient.