Using and Understanding Avalanche Photodiodes
For pont (2) The data sheet gives pinout.

AD500-8-S1 Datasheet

The third pin is the metal case of the diode. In general it should be connected to ground.

There will be a small amount of capacitative coupling to the photodiode itself (not specified in the data sheet) so the ground connection should be clean as any noise could couple to the active electrodes.

An APD needs to be biased at a high stable voltage close to its breakdown so that the photo-generated carriers are multiplied by avalanche action. For this diode it can result in an amplification of up to 1000 times.

The actual bias required is shown in the data-sheet and will probably require adjustment for each diode and may need temperature compensation.