Kazuya being more open in evilness during the time he's the head of Mishima Zaibatsu can have more meaning than just plain "He's probably eviler than Heihachi"; if you recount that Kazuya probably hates everything about Heihachi, so he probably held a disdain in Heihachi's practices about being a Miser, looking good in public while scheming behind their backs, so he decided to be more open in the evilness of the Zaibatsu, at least he's being honest that he's an evil prick and the Zaibatsu is an evil corporation, not a generous one. Not that it lessens his evil value, but it does shed a deeper motive for being openly evil.
In Tag 2 Jun is cold with just about everyone, but one notable exception is Nina. They might fist bump, which is more than what everyone bar her son gets with Jun. But Nina is Jin's bodyguard (and also one of the few partners that he likes), meaning she and Jun can find some common ground. For bonus points Jun's new win pose is similar to Nina's. Are we sure she's not evil?
In Tag, the first entry in the Tag series, Jun did not have good Netsu allegiance with Nina, either. Though by that point, for all we knew, in the third tournament Nina was an unwilling enemy of Jin's, having been brainwashed by Ogre to kill him.
Yoshimitsu's sword, according to 6, is possessed by a spirit that craves the blood of evildoers. His sworn enemy is Bryan — a cyborg. As shown in the former's TTT2 ending, Bryan doesn't bleed when cut through by Yoshi — he explodes instead. In other words, even if Yoshimitsu were to defeat Bryan, his sword's bloodlust will not be satisfied and will still attempt to eventually possess him. No wonder Yoshimitsu is very, very pissed off at Bryan. That said, Fumaken (the second sword he picked up in 6 to seal away the cursed properties of the Yoshimitsu blade) is supposed to mitigate all of that. Yoshimitsu's loathing for Bryan is simply due to what Bryan did to Doctor Bosconovitch and his fellow Manji ninjas prior to the fifth tournament. Anyone would take offense to that, and while Yoshimitsu is eccentric, he already doesn't take kindly to villains to begin with.
Jin winning the 3rd tournament despite Paul beating Ogre and Hwoarang facing Jin to a tie? Look at the structure for the earlier games before the number of characters really took off. You faced every other fighter except the sub-bosses, then your personal sub boss, (or in T3, just Heihachi), then the boss. The tournament isn't a one-and-done tourament, it's a round robin event. Ogre showed up in the middle of T3 and Paul somehow overcame him, and left assuming he'd won, because.... he's Paul. Jin and Hwo drew, but Jin still won overall (we can assume Paul left after he'd lost to Jin but not Hwo, so Jin still had more actual wins).
Marshall Law and Paul Phoenix being homages to Bruce Lee and Chuck Norris makes more sense if you know that the two were friends in real life, leading to Bruce inviting Chuck to appear in his film Way of the Dragon. So, them being homages to Bruce and Chuck also means that their friendship was a homage to Bruce and Chuck's friendship before Bruce died a year after the film's release.
If you compare Baek's movelist in earlier games, say, Tekken Tag Tournament, to his latest appearance in Tekken Tag Tournament 2, you can see that while Baek may have learned some moves, he's lost quite a bit too, unlike other fighter who typically get more complex movelists than they had in earlier games. For example, his Baek's special is missing in TTT2 and most other moves similar to his emblematic sequence of side kick x 2->Axe kick->Tornado kick->Spinning side kick, with the only remnant of this being his Tornado kick->spinning side kick in TTT2. This could be seen as an indicator of something most other fighters seem to be unaffected by: Baek is aging, and thus cannot perform some of his old moves as well as he could.
Now that the seventh game's story had been told, Jun's What the Hell, Hero? moment in getting her son to find Heihachi makes all the sense in the world. When he was young Kazumi tried to kill him many times and he brushed it off, seeing that she was mentally unwell and hoping that something would get through to her. It's not shown how much of this if any Jun would know but she persevered with Kazuya no matter how evil he was. Heihachi of all people would know a thing or two about raising the son of someone who came across as mentally unwell (due to, y'know, the Devil Gene) not to mention his own grandson.
Heihachi's taking over the family business of warmongering and continuing it seems like a Kick the Dog moment to his own father Jinpachi, but it makes sense; Heihachi is a man of martial arts and power, and was likely raised during the era of World War II or the immediate aftermath given he's in his mid-70's when he finally dies and ran the company for 40 years prior to the first game. Jinpachi had contributed military power and might during the war, only to grow to regret this decision, something that Heihachi probably didn't understand. He likely knew nothing but conflict as a driving goal as a result, and forcefully took the family company from Jinpachi to do what he thought the Mishimas were best at. This is also bolstered by Kazumi's betrayal that causes his Start of Darkness, as being forced to kill and take such a heavy hit upon his heart pushed him down the very path Jinpachi tried to step away from.