Front Cone Replacement Problem
 

Removed

If the new cones have flat spots cut for a cone spanner, then I think you will be OK to leave out the keyed washers because you can hold the cone with a spanner while tightening the lock nut. I have seen cones with no flats; they need the keyed washer so tightening the lock nut does not move the cone.

Typically axles protrude 5.5 mm/side but less is adequate if you tighten the qr skewer properly. For example 126 mm rear hubs (137 mm axle) are often respaced to 130mm (normally requiring a 141 mm axle) using the original axle with no problems.

However, you do want to maintain the 100 mm OLD dimension so the wheel will fit in the fork dropouts so reducing the thickness of the spacer washers is certainly a good option. As noted, if the new cones have wrench flats (and I've never seen those that don't, although they might exist) you can dispense with the washers completely.

GLad everything worked out. FYI, if you left out the keyed washer on just one side, then your rim will no longer be centered (when mounted on the bike)

But... and here is the rub....

Make very sure there is no grease between the cone and lock nut, you want these items to lock together and not budge on the road due to vibration. Before you fit them clean them with a grease cutting fluid. Mosy washers are ribbed on the flat to make then grip, if you leave out this washer you MUST have a clean surface tightening on a clean surface.