Dayve 10mm, as stated in the product specs. I measured mine to double check.

Dayve You probably could. You would just have to connect it to either the 3-pin input or the 3.5mm input. You may need a separate power supply for the Bluetooth module unless you can find a module that uses the same voltage as your using for the amplifier board.

Dayve It is a tally counter. Each time you press the big button, the number increments by 1. You could use it to count whatever you want. It has 5 separate counters so you can keep track of 5 different things at once.

Dayve 50cm (19.7 inches)

Dayve I'm just ordering one of these right now, so I can't say for sure. Based on the pictures, I assume that's where you would connect your circuit(s) to be switched. The two middle pins are negative (and probably connected together). The outer pins would be positive for circuit 1 and circuit 2.