how were copies of vinyl made in "third-party" countries

Pressing vinyl is pretty much a 3rd world process. It's not that tricky. The process is Masters, Mothers, and Stampers. The Masters are cut on a lathe using audio tape as the source in the old days. 

...if it's not that tricky why are there so many versions rated so differently?

If you run the Stampers or Mothers too long, don't watch your vinyl temps carefully, or run substandard vinyl, quality can suffer. Cutting the Masters is done early in the process and unless it's a bootleg, it's as good as it gets. 

Also, could they have also sent the mothers, stampers?

I could be wrong, but the master tapes are not going anywhere. The master disc is made from a lathe cut lacquer and they can make a few masters if needed. I think I said it wrong earlier. The record stamper company would get a master disc. I wasn't in the industry so if I'm wrong I'm willing to be corrected. 

...and the entire mastering to lacquer might take place at each pressing plant, or just some pressing plants. Again, depends.

A lot depended on how many units they thought would sell. If caught short they may run the stampers longer or make new masters.

is there a way to tell if an LP is likely to sound better from looking at it? 

The answer is no but I always looked for coarse grooves. That shows they were changing the pitch for more dynamic range but there are good sounding records that show even spacing also. 

my 90s Hungarian LPs are absolute crap while 70s and 80s are delightful. So much for capitalism :)

I wouldn't condemn an entire economic system based on a few bad LPs.  :)

@russ69 I used the smiley, I was joking. 

Me too!