I'll take a stab at this. There are a lot of variables here and without any other parameters other than stated it's difficult to give a definitive answer.
A lathe with live tooling and a C axis or milling machine would work. A milling machine might give better tool path options due to the CAM software available.a) Which type of machine (mill, lathe, other) is appropriate for each type of cut?
A rotating tool would be what I would use. Due to the changing radius of a helix, a fixed grooving tool would not work well.b) What type of tool (rotating side cutter or end cutter; lathe parting-tool type; other)?
I have slotted steel with a 1mm endmill. Requires a rigid machine and good control.c) How thin can I make the slot?
At least 5x the diameter of the cutter, maybe deeper with some care and ideal conditions.d) How deep can I make the slot?