Wheel shouldn't be dished and doesn't look dished from your photo but probably can't tell very well from that one photo. Do you know how to tell if it's dished yourself?
Putting that aside for just now though, are the spacers/locknuts etc on either side of the hub the same? They should be. They will need to be.
Your fixed sprocket should look like this
with the side which is facing up in that photo against the hub, that should put the teeth in much the same position relative to the hub flange (and the shoulder at the bottom of the threading) as the freewheel side.
If your cog is the correct shape and fitted the correct way around and the spacers and locknuts are the same on both sides of the hub then the chainline should be the same whichever way around you have the wheel and you then just have to match the front to that.
That means that at most you'll need to replace some washers/locknuts (so that your hub sits centrally in your frame), your cog (so that the teeth are in the same position as the freewheel teeth)and your bottom bracket (so that the chainring lines up with the freewheel/cog). Even if your cranks are road cranks and your chainring is not currently sitting in the right place you should be able to correct that with some combination of bb axle length and chainring on either the inside or outside of the spider