Do a voltage check, check resistors and fuses. Or post a photo of mainboard.

Check your cart connector.  I expect there's some solder debris down in there that's sometimes bridging the pins.  Maybe spray it out with canned air or sweep it with a small brush/pipe cleaner.

^ Yes, I did. It doesn't break anywhere.

I don't think that the problem is in display, because if it was, the ROM would boot fine (i'd be to make sound with muddygb by pushing d-pad).
Also, as i've already mentioned, it somewhy started working well, but then glitched again.

Did you check the LCD ribbon cable?

An attempt to mod my dmg have led to unability to start it up.

Well, after finishing the mod (and, afterwards, removing it) I couldn't start my dmg.
When I was trying to boot lsdj, the startup was ok, (Nintendo logo falling to the screen center and makes bleep) But the lsdj didn't start. Instead, a black screen appeared.
Booting smaller ROMs (muddyGB) didn't work too, the loading have been freezing after the bleep.

Later, i've noticed some gray shit (probably oxide) on contact pads inside the gameboy.
After cleaning it up, the dmg started well, with both ROMs. lsdj save file was corrupted, but not destroyed completely.

On the next day, it stopped starting up again.
Now, booting lsdj showed some pixel pattern. Booting muddyGB was the same. After a few on/off cycles the startup logo started flashing, if the cartridge was inside.
Otherwise, this ↓ happens.

The logo stays the same most of the time, but sometimes it changes to a 'regular' rectangle, changes back, makes a horisontal line or morphs randomly.

What could cause the problem and is there a way to fix it?