Finally the problem was identified to be with loose ribbon cable from the power connector (the plate with the ON switch) to the MB which runs above the fan. It is narrow - probably around 5 leads and if someone is moving the laptop often, because of he plastic body twisting and turning, it can get a bit loose and there'll be no proper contact.

Another thing - the MB side under the keyboard (i.e. the one which is accessible only if the whole MB is disasembled) accumulated an unbelievable amount of fine dust. Had a hard time to clean that, it is amazing that it did not short-circuit the whole thing. 

Anyway, now the machine runs for a second day without any visible problems, except that when performing even light tasks the fan is running most of the time at high rpm. I have changed the thermal paste, its probably because of the hot weather, but did not witness such behavious so far. Maybe the newly installed Windows 10, even though it's the 32-bit version, is a bit too much for this machine? Or because there are constant updates going on all the time.

Thanks! However, it turns out it's not over yet. I brought it home with the intention to give it to my mother to keep in touch with the world, but the shutdowns started all over again. I am sure it's the loose connection again. Now, I do not think I need to replace the ribbon cable (testing it with multimeter generally says it's OK), so it's the contact with the connectors and when reinserted  - it works for a while. Hopefully the connectors themselves will not be the problem, as it's more difficult to do replace them. Anyway, will try to put some silicon glue to harden all over the connector and fix cable and connector in one indivisible part and see what happens. It could not become worse, anyway. Actually, the whole problem started when after all this years on the desk, I started to carry the laptop on a couple of occasions to seminars. It just has too much plastic and is too flexible, in order to have the sufficient structural strength as needed for a portable device, that's my opinion.

Anyway, after all this exercise, whether it is over or not, I can probably qualify as good as anybody as an Acer hardware support, especially given the level of it here, even though I have only some basic technical background :-) 

As regards the software, I think the intensive work of the fan (and processor) was due to the machine being offline for the past month or so and now did almost 2 full days of whatever updates MS threw at it. Probably should have attempted clean install and not keep settings and documents because I have them elsewhere anyway. I would consider that now the registry file has swollen as pornstar boobs :-) Only 2 Gigs of RAM do not help it either even though it's the 32-bit version. Its DDR3, so it won't be so expensive compared to DDR2 to increase it when everything else is OK. Then will try it with 64-bit version as in my experience it is more stable and non-problematic in respect to drivers especially the latest releases.

Speaking of drivers, another thing is that MS installed as drivers whatever they consider certified and tested for legacy hardware, but the reality would require that I go to hardware manufacturer websites and download the latest drivers for every piece of hardware, even though it may be in some cases for earlier versions of MS Windows. 

For now I am unwilling to do that, because on a previous instance when I used driver search and update programme (Iobit Driver Booster or something like that) for a legacy laptop updated to Windows 10, devices like the display, touchpad, USB ports refused to work with their suggested manufacturer's drivers and had to revert to MS suggested ones for Windows 10. Anyway, that was some time ago and by now manufacturers probably have done some effort to supply proper drivers for Windows 10 even for older devices or at least I am ever hopeful.

Anyway, I will not downgrade to Win 8.1 or 7, even if I have to buy cooler pads for older laptops :-) The step is made, Rubicon is crossed, there is no way back :-) In this field - software and hardware, things quickly become obsolete and better to get used to the new, than to avoid the change. I am aware Apollo spaceships were designed and launched with computers with the capacity of a modern calculator and OS like the DOS. Also probably the most used OS worldwide is still the Win XP, or maybe 7, but be it solely for reasons of information security, better to use the latest OS, no matter what the theorists of conspiracy say about it :-)