As the title says, can I select Basic Weapon Training (Universal) or Pistol Weapon Training (Universal) as a part of this 'Trials and Travails' origin path or must I select something like (Basic Weapon Training (Las) or Pistol Weapon Training (Plasma)?

It seems that in RT training in the individual weapon subtypes is uncommon with such options being greatly reduced - such as with Melee Weapons Training - but as all of the X Weapon Training (Universal) Talents usually cost 500 xp, this makes The Hand of War appear to be too generous of a deal.

According to the RAW, I guess there's nothing preventing you from taking it. However I get this strong feeling that this will be rectified in the first iteration of a Rogue Trader Errata...

Varnias has it right, I think. Universal training talents seem to be something that would only become available once you reach Rogue Trader level.