When adding the 4mm, if you add it equally on each side no redishing is needed to center the rim. Most hubs for 6 speed freewheels are not highly dished, so trying to strengthen the wheel is probably not needed, especially if you don't intend to do anything extreme with the bike or are not too heavy. Unless the wheel currently is out of true, spoke adjustment won't be needed. Depending on the current setup of spacers and locknut with, you may be able to swap parts between sides and make up the differences as needed.

Generally axles have 5-6mm outside the locknuts, so adding 2mm to each side should not causes any problems. 3mm on each side is plenty, especially on steel dropouts.

With the addition of 2mm on the drive side, a 7 speed freewheel should fit.

The shifter speed needs to match the # of cogs. Newer cogs have profiled teeth and engineered shift points that are much improved.