How was the KGB supposed to do anything when the politics were dominated by authoritarian, self serving and very suspicious leaders such as the likes of Arafat?
The way they did everything else-- diplomatic and economic pressure, carrots and sticks, assassinations and misinformation.
The KGB was quite good at these things and knew well how to manipulate leaders and organizations once they had their hooks in deep enough. They were the premier intelligence service of their time.
When the PDPA was having factional feuds (and was also "dominated by authortiarian, self-serving and very suspicious leaders"), the KGB solved that through a swift, bloody purge of the most problematic people within the PDPA and then leaned on the factions through diplomatic, military and economic pressure to re-merge. That the PDPA was eventually overthrown does not change the fact that the KGB was able to severely reduce factionalism and disorganization within the PDPA.
The inter feuding would have made organisation impossible.
See above. Difficult, but not impossible. In the example above the KGB had to forcibly merge the two warring factions of the PDPA on two separate occasions, but when they did the organizational consolidation reaped immediate rewards (the first time directly resulting in the PDPA's seizure of state power).
Soviet training was also not up to it
Sure they were.
and there were quite a lot of complaints about the recruits.
Yeah, but with good organization, weaponry, training and a well-developed logistical base, those problems can be overcome.
The Arabs were just as weary of the USSR as they were of the US and each other.
I presume you meant "wary", well this varied from country to country, leader to leader-- certainly Nasser developed a very close relationship with the Soviets despite being one of the leaders of the NAM. In any event, wary or not, if Arafat (or any other Palestinian leader) were offered the KGB's assistance in consolidating their power, it's likely they would have taken it. But that shit doesn't happen overnight-- it's not like the KGB drops in out of the blue and says "Hey, we can help you"-- it requires very careful cultivation of contacts and assets within an organization, but the KGB never developed that level of infiltration of the PLO.
Besides the USSR were far more interested in their issues rather than finding a solution.
Well, sure, but if they could have Palestine as a reliable client state, and knocked out the West's main military outpost in the region (Israel), you don't think they would have taken the opportunity? History suggests they would have.
There's a reason it didn't happen (well several reasons), so ultimately you are correct about the obstacles, mistrust, and factionalism/infighting in that region being too great to overcome as it happened, but what I'm suggesting could have happened
isn't that far out there either. If I had a time machine and the trust of Soviet intelligence, knowing what we know now, I'm sure I could have done a better job of it at least.