Neck shape...

Is the neck shape consistent for the various Les Paul models built in a given year? For example, would a 1959 Custom have a similar neck profile to a 1959 Standard or for that matter, a 1959 Junior? I understand that the neck shape can vary between two guitars from the same period but allowing for that, were the general dimensions more or less similar across the entire range (leaving aside 3/4 size guitars and so on)

No. Not just types but each individual guitar is unique. There are broad types, but any particular guitar may not conform to the average-

I tried out a couple of 56 goldtops and none had the same neckshape. Pretty interesting. It's the samt thing with stratocasters.