Black screen on start, then restart and okay
 

1) Finally, my teenage Acer Desktop (Pentium dual-core E5400 @ 2.70GHz × 2 CPU with Linux Mint 18.2 Sonya 32-bit, generic Kernel 4.15.0-48-i686) has a problem.
When I turn it on nothing happens, on the black screen you can only see a couple of light lines. But if I turn it off (by pressing the power button for 5 seconds) and restart, everything is fine. I mean, I see the bios options and then the system loads normally.
2) Where can I start to find a solution?
3) Should I buy a new one? I'm stingy but realistic ...

I don’t consider this a huge problem, since you can boot your PC, but yes, not seeing the BIOS screen does point to some hardware flakiness.

Personally I wouldn’t bother. Since you seem to worry though: I empathize with your stinginess (only one of my numerous computers was bought new, an HP microserver that cost me the incredible sum of ~130 US$), but Ebay is awash with compute equipment, so what are you waiting for? 100$ should be sufficient to upgrade to a 64 bit laptop with 8GB memory and a decent CPU.

some people don't see video for many seconds (ie, until after linux messages begin printing). i would not reboot just for that.

sounds like spring cleaning

#1 dust kills electronics. unplug everything wait 30 sec take it outside blow out the dust. DO NOT blow or brush the fans unless you plan to replace them

#2 chip creep. reseat any removable chips

#3 video card cables. your video card is WEAK so your cable needs to be good (clean, plug and unplug it). that will cause video detection lag during booting, yes. a new video card, new led monitor or both is a different way to address the issue.

that's where i would start. i just ignore it - but for me i get video on several seconds later so i don't bother rebooting. so far.

https://www.tomshardware.com

great site for articles on hardware artifacts like slow video detection during boot

I DIDN'T SEE at first you said "solid line". that could be a video card problem.

however before jumping to that conclusion i MUST ask if you see ANY vga/text from bios or bootloader or boot messages BEFORE you see the dark - why is because it very well could be a naughty linux OS distro who has messed up your video card (bad save/restore video state commands behind X or Wayland or other anti-X solution)

I cleaned the whole motherboard (a lot of work, if done accurately!) and now everything works.
Thank you all.

I always use a household vacuum cleaner.
Never killed any component by electrostatic death - yet!

[This is a risky procedure]