The idea of using things like insight to see if someone is about to move to violence and the like I'm cool with.
However, some of the other ideas here hit into three concepts that do not work well with 5e as written. To some point these can be addressed by rule changes.
5e uses the concept of "until your next turn" as shorthand for "affects each target for one action". If you change initiative by round, this gets thrown off. Some can be affected for for multiple turns (say going after in one round and before in the next), other might suffer no effects at all (going before in one round and after in the next).
Side initiative can lead to really overwhelming focus fire. Because there's no repositioning between different moves or chance to heal, it really leads to people dropping with less chance to be tactical. Neither of those are positive enjoyment factors for players.
Lastly choosing order among players, such as popcorn, really changes the value of some abilities. For example, it makes a mockery of getting dropped to zero - when you can make sure you will never lose an action because you will always get healed before it's your initiative, but not to far in advance you might drop again, it exasperates the "pop-up healing" issues tenfold. Other things like pushing a foe prone could mean that your then have a bunch of that side able to attack with advantage without any chance of the target getting a move to get up. Heck, do it late in a round and you can have all of the melee members of your party go twice - once this round and then at the top of next round - without any chance for the foe to get up. And of course the other way works with intelligent foes.