Prohibited sign at boot

First of all, let me clarify that I already did quite an extensive search on this "prohibited sign at boot" problem, but couldn't find anyone with quite the same difficulties as I am facing.

I was having some serious problems with my Macbook Pro in the past few months, mainly when using Skype. Almost every action related to the app would give me the Spinning Beach Ball, from initialising it to picking up or ending a call. I constantly found myself having to Force Quit and reopen the app.

But this I could bear it (not without frustration, granted). Until one day, during a video call plus (perhaps) intensive web-browsing, the Beach Ball became more constant and would take longer to resolve, culminating in a total crash and the system forcefully rebooting itself!

To my surprise, after the reboot I was faced with the Folder With Question Mark sign, which sometimes flipped to a Prohibited Sign, sometimes quickly became an Apple Logo and back to the Folder sign, and so on.

I tried to boot into Recovery Mode, checked the disk with Disk Utility (results were fine), tried to reset the Startup Disk...nothing seemed to work. I sort of gave up and left it aside for a few minutes, after which I decided to give it a last chance and reboot it once more...and it magically worked!!!

Having strongly suspected that the OS had been somehow corrupted, I decided to format the HD (!!!) and do a fresh install. Much to my surprise, the prohibited sign at boot CONTINUED!!! I couldn't boot a freshly installed OS!

However, there was a trick: again, if I shut it down and waited for about 5 minutes, it would then boot normally. And work as if nothing had ever happened. Except, again, for Skype, which continued crashing the whole system.

I tried PRAM reset. Doesn't seem to work. Did NOT try SMC reset yet. Could it be useful?

To summarise, this is the problem I'm having:

Prohibited sign at boot, which does not go away even after reinstalling the whole system.

Problem seems to be "solved" by shutting laptop down, waiting for a couple of minutes, and booting again.

I suspected HD failure, but when the system does boot, it works perfectly (again, except for Skype). So I don't want to buy a new HD just to see the problem continue. Also, Disk Utility would return perfectly OK results.

I also suspected overheating issues, but couldn't find a FREE up-to-date temperature monitor. And sometimes the problem occurred after I restarted the system having used the laptop for only 10 minutes (when I was updating and installing software in my newly-installed OS), and the laptop felt quite cold still. (But, I know, "feeling" is no good measure here).

Any help is appreciated. I don't want to spend loads of money on a new laptop if I can still save (and trust!) this one, which, to a great extent, seems to be still functional.

Thanks a bunch!

Specs:

Macbook Pro mid-2012

4 GB memory, 500 GB HD

OSX: problem originally occurred with El Capitan. Re-installed Mountain Lion, and problem persisted. Now running El Capitan again, OS 10.11.6.