How can I widen my nut slots without files?
As for the nut, one quick way to widen the slots if you don't have special nut files is the use the bit of string you cut off from the locking tuners and use it as a file. Just run it through the corresponding slot to make it bigger. Go slow so you don't make the slot deeper, just wider.
How deep should nut slots be?
The slots should be only deep enough so 1/3 of the string is above the top of the nut and 2/3 in the nut.