Q1: Drill bits. The hole required is either 22mm or 25mm (depends on the kit). Flat drill bits of this size (suitable for wood or soft plastic) are readily available. I used these but I suppose you could file the hole larger from the larger diamter bit you have, using a rasp.
Q2: My downpipes also go into the ground, with a neoprene cover sealing the bottom of the downpipe which sits over the top of the (plastic) 100mm soil pipe that fors the soakaway. You have to remove the pipe to be able to cut a section (typically about 25mm) out of the pipe in the position where the diverter unit sits. That is the way to avoid lowering the bottom of the downpipe.