Rolf,
It doesn't need to be more than about a 25 thousands of an inch ( 0.63mm ) deep. That's deep enough to make the undercuts without upsetting the metal on the surface if your're careful. If you cut the groove too deep, you'll have difficulty getting a graver in there to make the undercuts without maring the top edges.
If you cut 0.025 deep, use a wire about 0.040" ( 1.02mm ) diameter for the inlay. You could use a somewhat smaller wire, but 0.040" will fill the undercut and groove, and leave you some material to file off from the surface.
You only need to undercut about 0.010" ( 0.25mm ) or so, and for a 1/16" wide groove, don't really need to raise burrs, but certainly can if you want to be double sure the inlay stays in place.
John