The way they treated the Vulcan DNA is stupid as fuck, of course. Vulcans work to suppress their emotions, they aren't born as completely emotionless robots (the episode even acknowledges this by making La'an different.) Injecting "Vulcan DNA" into you wouldn't just make you turn into a perfectly logical asshole with Vulcan hair instantly. And - and I'm being too serious here I know
While acknowledging that it's just an excuse for hi-jinks, the treatment is based on an incredibly advanced technique that they can't fully understand. If that includes installing brain training (which the episode explicitly says carried over from Spock), that's within the rules they set.
There's no "dictator gene" to be passed on.
Personality traits are definitely passed down genetically in families, separate from "nurture".