Are current audio cable designs superior to 1980s designs?

As long as your connectors or wire isn't starting to oxidize they're fine. If you think you need new speaker wire use something like canare 4s11, for interconnects monoprice has some good ones. 

YES!!!

You asked a generic question, followed by several more focused questions related to your specific cables.  To your generic question; it depends on your definition of “superior.”  Connectors have improved so superior in that way, yes. Otherwise, if you mean sonically and audibly, then probably yes also but maybe not in the multitude of ways or to the magnitude that cable manufacturers would have you believe.  With power transmission (i.e., power cords and speaker cables), wire gauge seems to be the main factor with maybe some audible effects between solid core vs. stranded wire.  With small signal cables, wire gauge, geometry and shielding seem to be key factors, but you don’t need to pay a lot to get those benefits.  Many believe relatively pedestrian Mogami balanced cables sound just fine.Other stuff such as copper purity (beyond OFC), cryogenic treatment, wire direction, exotic wire shapes, exotic dielectrics, etc. seem to be audible to some but not all.  Even if you hear differences, are those differences significant, are they really improvements, and are they worth the many thousands of dollars cable manufacturers have charged over the years?  You need to listen and decide for yourself.  I went down that road for awhile, wasted a bunch of time and money I will never get back, and decided I would rather enjoy the music than worry about cables.

I'm just kind of impressed that some audio manufacturer was putting XLR jacks on the back of their amps in 1987.  Preamp AND amplifier, who was this manufacturer?  

I really hope it wasn't a pro amp like Crown or Belringer.

Preamp was Bryston BP series; amp was B&K 4420, dual mono in single chassis.Regular consumer grade components.