The spacers go on the axle. It's to make inner part wide enough to fill the space between the dropouts. You don't do anything with the cog; that would be weird.

I'm guessing that the chainline is wider at the chainring than at the cog. Take a look at your BB spindle. A lot of old road bikes have offset spindles so that you can make it shorter/longer by flipping it around. I've had to flip the spindle on both of my gf's bikes (an old Peugeot and a Fuji Monteray) in order to get a decent chainline. We had to put the chainring on the inside of the crank spider for the Peugeot as well, so you've probably still got options after you re-space the axle.