It's of course impossible in practice, but certainly not in the context of a thought experiment such as this. But it doesn't matter too much: the main takeaway here is that you never make a mistake if you assume that the Brownian dust particle has a well-defined position at all times. With electrons that assumption leads to crassly incorrect predictions.

Yes, but it does move; the only thing that gets reset upon measurement of the Brownian mote are your predictions. With an electron, future measurements get reset as well.

That's the (in)famous collapse of the wavefunction. You can do an experiment yourself with polarizing filters: put two filters at 90 degrees with one another, no light goes through. Put a third filter at a 45 degree angle with both, and now some light goes through. That can be understood classically without much difficulty, of course, but the quantum perspective is illustrative: each filter corresponds to a measurement of photon polarization, and every photon that passes will be polarized along the direction of the filter. It turns out that this much is completely general: if you make a measurement, and get a result, an identical measurement immediately following the first must give the exact same result. It's one of the postulates of quantum mechanics, albeit the most controversial.

A textbook that is unusually clear about these points (in my opinion) is Ballentine's, particularly chapter 9.

Scott Aaronson's series "Quantum Computing since Democritus" is fantastic reading in its entirety, but in this context I especially recommend this page:

https://www.scottaaronson.com/democritus/lec11.html

The relevant section is titled "Invariance under Time-Slicings".

Note I never said anything about a "conscious being". To my knowledge quantum mechanics doesn't say anything useful or meaningful about consciousness. It does tell you, as in, a creature or artificial intelligence sophisticated enough to perform measurements, what results you should expect. And it turns out that it what you know (or, more properly, what you could possibly know, given the information available to you) does affect future results.

So, if I'm understanding you correctly, your proposal is that the photon is some object akin to a classical wave, and that running it through a detector introduces some uncontrolled phase error that shifts the interference around in such a way that no pattern emerges after many trials? It's an appealing, plausible, reasonable sounding suggestion, but it doesn't work: one clear example of why it doesn't work is given by the delayed-choice quantum eraser experiment.

In this (widely misunderstood) experiment, you start with the two slits and single photons as usual, only after the photon goes through the slits it enters a nonlinear crystal that splits it into an entangled pair. One half of the pair (termed the "signal" photon) gets taken to a movable detector, which takes the role of the screen and records detection events (which will evince, or not, an interference pattern). The other half (termed the "idler" photon) goes through a set of mirrors and splitters, eventually hitting one of a set of 4 detectors. If it hits detectors 1 or 2, it could've come from either slit. If it hits detectors 3 or 4, it could only have come from slit B or A respectively. So an idler detection event at one set of detectors gives which-slit information (which can be used to reason about what the idler photons will do), and a detection event at the other set does not.

Important: the path taken by the idler photons is much longer than that taken by the signal photon, so the signal photon is detected first, then the idler.

You let the experiment run, accumulating detection events at both signal and idler detectors. You can now tally up these events based on what detector the idler photons hit, and what you find is that, with the cases where idler gives you "which slit" information you don't get an interference pattern with the signal photons, but if you tally up the events where you don't know where the idler came from -- when it hits detectors 1 or 2 -- you do get an interference pattern! The detection of the idler photon can't possibly have disturbed the phase of the signal photon in any way -- it's a different particle, far away, and it's already been detected!

What happens is something more subtle. With any detection event, by definition, the detector must get entangled with the detectee. "Entanglement" means that you no longer get to ignore the detector if you want a complete statistical description of what will happen, and in particular, if you want see interference effects, you have to put the entire detector in a coherent superposition. This is de facto impossible with macroscopic objects, so what we see with the ordinary two-slit experiment is the utter destruction of the interference pattern upon detection. What the delayed-choice quantum eraser experiment does is mess up the coherence of the photon in a controlled way so that this can be (partly) undone and the interference pattern recovered.