How can you track the paths of these particles, without interfering
with the particle, or destroying it?
You don't. In fact, the paths are not "natural", they are being created as part of the measurement process.
Basically the detectors rely on the particles having so much energy that they will not be disturbed too much by the interactions with whatever they are using to detect them. There are any number of ways to do the detection itself, but most of them rely on some sort of hyper-sensitive state that the passage of a charged particle will upset.
For instance, in the bubble chamber, liquid hydrogen is put in a critical state. When a charged particle travels through it, the rapid passage of charge ionizes the hydrogen, which causes a bubble to form. The bubble is large enough that you can photograph it. The particle is slowed by the interaction, but that's ok.
Now we have to identify that particle. Normally the way that is accomplished is by putting the chamber within a really powerful magnet. The magnet will interact with the charged particle and cause it to curve around into a circle. The radius of that circle is defined by the charge (which is generally 1 in these cases) and the mass of the particle. So by measuring the radius you can figure out the mass and thus identify the particle.
Look at the first image you posted. Do you see a series of circles right near the middle? That's a single particle. When it was first split off, in the reaction in the center, it was travelling to the left. The magnet pulled it around into a clockwise circle. As it was moving in the circle it was reacting with whatever detector was being used, and losing energy. So that's why the circles get smaller and smaller.
You'll also notice lines that look perfectly straight or are very slightly curved. These are all curving, just so little that they look straight. That means the magnet is not effecting their path as much, which generally means they are heavier (as opposed to having less charge).
There's all sorts of actual detector systems, but generally they all work something like this. You can even use photographic film as the detector, which used to be quite common. There are systems that look for the tiny amount of electricity created when a particle passes, ones that use scintillation of a crystal, all sorts of different concepts, but in the end they all work basically the same way in trying to identify what's happening.
Best Answer
OK, here is something concrete and quantitative, "Guidelines for predicting single-event upsets in neutron environments":
Cosmic ray spallation.
From "Cosmic ray induced ionization in the atmosphere: Full modeling and practical applications":
It does not describe it, but the CORSIKA (Cosmic Ray Simulations for Kascade) simulation tool is used.
Google-fu:
quantitative neutrons cosmic rays atmosphere