Why mosfet drain is connected to its case?
This has to do with how a Power MOSFET is build up:



The Drain is at the bottom. This is convenient as the drain is also the location where the power is dissipated in such a device.

So this Drain can get hot and to cool it the heat must be conducted away from the device. This works best if the Drain has a large metal plate like the back of the MOSFET you mention.

At the same time it is a convenient way to fabricate the MOSFET this way. This has to do with doping levels and processing steps. Read the wikipedia page to learn more.

To cool a transistor you should mount it on a heatsink. If that heatsink can be electrically connected to the Drain of the MOSFET you have no problem.

If electrically connecting the Drain of the MOSFET to the heatsink is a problem then you need some form of electrical isolation. To maintain cooling capabilities you should use a material which does conduct the heat but does not conduct electricity.

Here's a practical example of that:


Note that the screws will connect to the heatsink so plastic rings might be needed for proper isolation.