> I'm curious, has any research been done on HOW they "choose" their state when observed?

They do not. There is no "particle choosing", ever.

What happens instead is that you (the observer) are also composed of particles. This is something that people are kinda willing to admit, but afterwards they deny the logical consequence... that the particles that create you can also be in more than one state at the same time.

What happens instead is "entanglement", which means that certain options of one particle become connected to certain options of another particle. All options continue to exist, but only some of the relations between different particles exist. Specifically, when observe the particle, the option "particle going left" becomes entangled with the option (remember, you are also a system of particles) "you see the particle going left"; and analogically, the option "particle going right" becomes entangled with the option "you see the particle going right".

Now each version of you observe one specific outcome... and can ask themselves 'what made the particle choose this specific outcome?' But there was no choice; it's just that the other movement of the particle is entangled with a different version of you.

Yes, parallel worlds -- the true shocking discovery of the quantum physics. Actually, all physicists admit that parallel worlds exist on the microscopic scale, the only controversy is that half of them say "and it's the same on the macroscopic scale, too", while the other half goes "I can't believe in macroscopic parallel worlds, therefore there must be a magical thing called 'quantum collapse' which makes this madness stop before it gets too large". (The belief in quantum collapse in unfalsifiable, because no matter how large system of particles existing in parallel configuration you build, the possible counter-argument goes "but if you make it even larger, then it will finally collapse". And this goalpost can be pushed indefinitely.)

Unlike in science fiction, the parallel worlds of quantum physics have the property that the more they differ, the less they can interact. If they only differ in position of one particle (one world where the particle goes left, another where it goes right, but everything else is exactly the same), there is a lot of interaction. But if they differ in hundreds of particles, the interaction is almost zero. The disappointing part is that you can never observe the parallel universe directly, because - again - you are composed of particles; even your very act of thinking means moving millions of particles in your brain around... so just the fact that you noticed a particle moving left has put millions of particles in your brain in a different position than if you would have noticed the particle moving right.

The problems with thinking about quantum physics arise when we imagine that the microscopic scale is just like the macroscopic scale, only smaller. ("Particles are like oranges, only smaller, duh. Actually, they are more like waves on the lake. Anyway, must be the former or the latter, because those are the only options I am familiar with.") When instead, the macro scale is a statistical aggregate of the micro scale, so we can't observe e.g. things that get cancelled out by other things. Instead of trying to fit the micro scale into our everyday experience, the proper way is to study the micro scale by its own laws, such as probabilities being expressed by complex numbers, and then think how could our everyday experience arise from this, i.e. again, what laws would then govern huge systems of particles, such as ourselves.