Schottky Diode Case Connection - Why?

This connection minimizes the thermal resistance between diode and case which improves cooling when mounted on a heatsink.
It has, unfortunately, the side effect that the heat sink becomes "live", too and that the heatsink creates unwanted short circuits between components mounted onto the same heatsink.
The remedy is isolated mounting using thermally conductive insulators and plastic screws or plastic sleeves to insulate metal screws. A well intelligible description is e.g. in this document.

 

Dual diodes have 2 diodes, single diodes have 1 diode. You use what you need.

 

The power dissipation is equal to P=I*V where I = current through the diode, V = voltage across the diode.
The current depends on the way the diode is used and which curent flows through it. The voltage you take from teh datasheet for teh given current. A Schottky has ~0.3 V voltage drop.
When using fast switching currents/voltages, add the dynamiv losses which are not that easy to calculate.

From the thermal resistance (given in K/W) of the heat sink and the diode and the power dissipated you can calculate the heating of the diode. Here's some background in case you are willing to dig through 36 pages of documentation

It depends on the power dissipation, see above. They can get hot.