either or all: (1) check the battery - settings app -> battery — under battery health -> Peak performance capability - it should say that battery is supporting normal peak performance. If not — a battery needs replacement. (2) some short inside or water damage and thus corrosion or some in-between so that if power usage exceeds the base levels - the battery does not cope and / or short occures and the system shuts down. (3) loose connector The phone may need to be opened to evaluate 2. and 3.

No need to " install the OS off a USB device (make bootable os installer on a USB device) - the already available and achieved OS net install will well do. If you got a spare external hard drive / SSD on a USB case or at least 16 or 32 GB USB stick -- you can install the said available macOS (mountain lion) and then proceed to boot from the freshly installed external. Once there the system profiler (system report in later os versions) will show whatever storage is recognized. I would have suggested starting with diagnostics (support.apple.com/en-us/HT202731 ) though the codes produced from the net diagnostics are so cryptic as to be barely usable - still one may get an answer that is usable in some cases. Additionally: The SSD module is Apple proprietary form though in essence - a SATA one - so a cheap adapter can be bought in eBay to transform the internal/proprietary into a SATA connected and thus usable in some external usb form. As usual - in any contacts - there are two sides being connected together...

The same here - Battery “need servicing” as per the battery health section.

Also - if the battery is depleted down so the phone shuts down, the phone starts twice upon starting the charge — ie plug in the phone in the charger — the phone starts up and then upon reaching the enter the passcode shuts down as if there is insufficient charge. in a moment it starts anew and completes the boot as supposed. not a big hassle - just annoying. Suspicion that the battery calibration is not happening as supposed.