This sounds like some power IC has a loose connection or cold solder joint and jostling and heating recreates a connection. To be certain, you could try booting with the laptop back panel open, if it fails, use a heat gun to gently apply heat to the motherboard and see if only doing that (without jostling) allows it to wake.
They hold “system binaries” meant for root user. It’s not a hard distinction but many if not most Linux fundamentals have their roots in very early computing, mainframes, Bell and Xerox, and this good idea has been carried into the here&now. Not sure about the provenance of this one, but it makes sense. isn’t /mnt /media different between distros? These aren’t hard and fast rules - some distros choose to keep files elsewhere from the “standard”.
/bin and /usr/bin, one is typically a symbolic link to another - they used to be stored on disks of different size, cost, and speed.
https://refspecs.linuxfoundation.org/FHS_3.0/fhs/ch03s16.html
https://unix.stackexchange.com/questions/5915/difference-between-bin-and-usr-bin