I'm Having Sudden Crashes Often Followed by an Automatic Reboot

[Gualicho32]

I'm currently running Mint 18 (Sarah) 64-Bit version on a MATE 1.14.1 desktop. The Linux Kernel currently installed is the 4.4.0-93-generic x86_64. My hardware is a nearly 10 year old Gateway SX2803 with a Pentium(R) Dual-Core CPU "E5800 @ 3.20GHz x 2".

About half the time, the crashes occur right after startup on my desktop user display after login. Other times, they will usually happen while streaming a video online. What could be causing these malfunctions? From the looks of things, it appears that other people are also having issues with Mint 18? Is it something specific to this version? I apologize in advance if this is a redundant troubleshooting topic. I posted this here because I was unsure if it was the hardware or the software.

[all41]

These things are almost always caused by hardware failure--often the power supply unit.
Some things you can do to narrow this down:
Run the memory test offered on the grub boot screen.
Unplug and reinsert the mother board cables to make sure you have good connects.
Make sure the processor heat sink fins are cleaned of dust bunnies.
Check if the processor heat sink paste has dried.
fwiw:
My sister in law was having shut down problems so I slid the tower out of it's hole and found the ac power plug
had fallen out--after plugging this back in the problems went away--so I surmise the cable was barely making contact
and fell out completely when I moved the tower.

.

[catweazel]

If the suiggestions made by @All41 fail to fix the issue, I recommend turning the machine on and pushing down on the motherboard. It could be a cracked link or copper trace. It could also be dying capacitors. In either case, I'm sad to say, your machine is about to become a boat anchor.

[jimallyn]

Mine was doing that. I replaced the power supply, and all is well now.

[Gualicho32]

These things are almost always caused by hardware failure--often the power supply unit.
Some things you can do to narrow this down:
Run the memory test offered on the grub boot screen.
Unplug and reinsert the mother board cables to make sure you have good connects.
Make sure the processor heat sink fins are cleaned of dust bunnies.
Check if the processor heat sink paste has dried.
fwiw:
My sister in law was having shut down problems so I slid the tower out of it's hole and found the ac power plug
had fallen out--after plugging this back in the problems went away--so I surmise the cable was barely making contact
and fell out completely when I moved the tower.

.

I haven't run any of your other suggested tests yet, but I did notice that the CPU heat sink paste had a sticky texture similar to fresh putty. There are no obvious signs of damage on the CPU itself. Even after I cleaned the heat sink fins (the radiator-like structure attached to the cooling fan I presume) the problem persisted.

[all41]

Well you have eliminated some question marks.
At this point I would substitute the power supply unit if you have an extra machine.
PSUs are inexpensive on ebay