There are special "phono" versions of the DIN to RCA adapters that have the required separate ground lead. Or you can just jam a suitable loose wire between the cases of the DIN connectors and connect the other end to the amp ground, at least you'll find out if a right adapter would remove the hum.
Don't bastardize the original cable, odds are that you will be getting an old B&O receiver in the near future
As to the other problem, it's quite probable that you have a stretched belt. Member Dillen (Martin) can probably get one for you if you can't get them locally (it doesn't need to be a B&O belt, just one of the right size. The long belts used in these old turntables are getting hard to find nowadays, though).
-mika