Do particles record every interaction they make with every other particle they encounter?

Yes, but they ‘record’ their interactions with everything they encounter. Each interaction imparts some impulse but the echo of that dissipates with time.
Particles are interacting with other particles megaoodles of times a second. I’d expect it would be practically impossible to distinguish the particular event you were attempting to “record” from the events before or after it.

Yes, if subatomic particles interact as in collide, they become quantum entangled. Two entangled particles somehow “know” instantly what happens to their entangled mate. Electrons have spin. They can be spin up, or spin down. This spin is not a constant, it is a waveform. In other words, the exact same electron vacillates rapidly between spin up and spin down. Strangely, if you measure and observe to see what it’s spin is, its waveform collapses. It assumes either spin up or spin down as it was measured, and it stays like that.

Now, if two electrons interact, their waveforms become quantum entangled. While they are entangled, you could carry one to the other side of the planet and observe its spin, and not only would its waveform collapse, it’s entangled twin’s waveform would collapse instantaneously. If the one you measured froze in spin up, it’s twin would freeze in spin down.

Computer scientists are already hard at work researching ways to use this to fashion a quantum computer where each bit takes up only a pair of electrons. If you find this fascinating enough to invest the time, listen to Stanford’s Leonard Susskind’s lecture. It’s about an hour and a half, but fascinating, in-depth info on quantum entanglement.