Unless you plan to use this wheel on a frame that is spaced at 130mm there is no reason to change the O.L.D. spacing of your hub for use with a 6 speed or even a 7 speed freewheel. 130mm would most likely be required if you went to an 8 speed freewheel but those are expensive and becoming increasing harder to find + there is an increased chance of breaking the axle due to the leverage forces being situated further away from the bearing supporting the axle on the drive side. The cost of the axle conversion plus an 8 speed freewheel would probably cost as much as a new wheel with a freehub designed to accept 8, 9. or 10 speeds.

Since you stated you are sticking with a 6sp freewheel system, all of what I stated above about 8sp freewheels is pretty moot, but unless you are looking to be able to use your wheel on multiple frames where one or more frame is spaced at 130mm there is no reason for this conversion other than for the sake of personal satisfaction of doing it in itself.

If you go ahead with this you should verify the width of each of the dropouts to figure into your calculation for axle length. You can buy longer than needed and cut/grind to the appropriate length but you can't add length to the axle.
And yes, you will need a 4mm spacer....or since you are staying with 6speed then you can use a 2mm spacer on each side...then check and correct dish as needed.

-j