I wouldn't even call what Mat & Perrin did wrong. Both of their enemies ARE already enemies, and both are invaders. Both could be said to be at war with Mat's & Perrin's side, so all they were doing was carrying out legitimate military operations, regardless of their personal motivations. Mat fought the Seanchan not so much because of the threat to Tuon, but because it was impossible to extricate his force from Seanchan lands. He would have been perfectly happy sneaking out, except there was no way out with the army blocking the way into Andor. He had to attack the other Seanchan forces to lure that army into moving out of his way. The threat to Tuon was only brought up to eliminate the idea of using her to talk their way through. After all, armies are not as much of a threat if you have their ruler as a potential hostage, or a companion who is willing to grant you safe passage. That was not a choice for Mat, because they were looking to kill Tuon.

As for Perrin's campaign against the Shaido, they had his wife. While that might seem like a personal and private concern to modern eyes, in a worl like WoT, familial relationships are not only a valid basis for policy decisions, they are EXPECTED to be such a basis. That is why rulers and nobles marry their children off to form alliances - it is expected that your daughter's husband or son's wife will exercise his or her power in the interests of the extended family, or that their heirs will act to the benefit of their grandparents. In addition, the subjects would expect such actions of their rulers, too. If Perrin won't do all in his power to save his wife, the woman he sleeps with and is supposed to cleave to as one flesh, how can they trust he will protect near-strangers as some of his subjects are? If he won't go all out to rescue his wife, how far can an ally like Berelain trust him? A man who won't come to the aid of his wife when she is kidnapped is not likely to defend his allies or vassals from invasion. And, let us not forget, the Shaido took Alliandre as well. That was half of Perrin's mission in the first place, and aruably the more important half. Since it was not jeopardizing the other half of the mission, he was OBLIGATED to do whatever it took to rescue Alliandre.

The fact of the personal motivations that existed for Mat & Perrin do not make their actions any less justified. As Faile pointed out to Perrin in the Two Rivers, having a personal reason does not invalidate a righteous action. In that case, he was concerned that his private motivation for revenge caused him to unworthily get his friends killed, but as she said, killing Trollocs is a good thing, regardless of whatever additional reasons Perrin brought to the table. The fact that in a hypothetical choice he never had to make, Perrin thinks he would let Alliandre die to save Faile does not change the fact that his campaign to save both of them was not only a justified, but a mandated and necessary act.

Except in his defense, the Seanchan invasion was unforeseen, the rebellions in Tear & Cairhien collapsed without causing Rand any more bloodshed (as Egwene notes, that makes reconciliation afterwards exponentially harder, and it is far more true for real politics than should be true for Aes Sedai), because he WAS distracted with other matters and could not spare the time to send men in to die while crushing Toram, Darlin et al. As far as the notion of his imitating the Seanchan, it is a little more complex than that. The Seanchan have institutions and set procedures and have practically made a science of conquering and expanding their empire, and their entire military and bureaucracy is on the same page and knows their principles and practices. We see in tGS that Rand has to negotiate with his best warriors and play semantics to get them to go along with his tactics. He simply does not have the time, or arguably, the means, to thoroughly suppress resistance and incorporate the nations into a single polity, much less convert them to his subjects. The nations of WoT have moved beyond monarchism to nationalism. They identify as Andorans or Cairhienin or Tairen, and the Murandians who prefer their local lords or communities to their national identity are viewed as abberrant and backwards, much like a contemporary person who might say "I am not an American, I am a New Jerseyite" or "I am not English, I am a Yorkshire man." A millenium of tradition is a lot to overcome in time to face a crisis less than two years away. That is why Rand would prefer to work through trustworthy rulers like Darlin or Elayne, who know, understand and personally get along with him. That way the people's natural loyalties are maintained intact with little fighting, and the presumptive loyalty of the ruler is such that there is no difference in the people being loyal to the ruler rather than the Dragon Reborn.

Or one third, as the case may be.
That would be an interesting scenario, but with the set up of WoT, it is completely counter-indicated in this case. While it took bloodshed to put Elayne on the throne, can you really say that there would have been less if he tried it any other way? Unless the books had gone completely differently, there seems no way for a scenario to arise where Rand could get away with deposing Elayne in favor of anyone else. As it is, if Elayne can win Andor's hearts and minds, their relationship, if handled the right way, can be the vehicle to make Andorans accept Rand. Your proposed scenario could only work if they liked Rand but had turned against Elayne.
For example, perhaps without the advent of Rahvin, if the populace stayed hostile to Morgase and the Tower, and Rand helped them in some way so as to be popular with the common folk as in Illian, then when Morgase dies of VD or something, the Tower tries to peremptorily have Elayne crown while making some sort of PR blunder that turns Andor against them...sure. Then your scenario would be plausible AND be able to provide an interesting character conflict.