Please offer some advice.

But how can it be that in 2017, an old game can't run properly at a resolution of at least 1280x1024? I've already tried everything...

Windows 7.
Processor - Athlon X4 Quad-core 651 3.0 Ghz
Video card - Geforce 550 TI 1gb.
RAM - 8 gb.
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It certainly can. For example, on a wide screen with a resolution of 1680x1050, it's impossible to play at 1280x1024 - you have to set it to 1024x768. New doesn't always mean better. The game is adapted for older hardware and systems, and on newer ones, it may behave somewhat inadequately. You can, of course, tinker with the settings (and the effect may come from a setting that doesn't seem to have anything to do with it), but there's no guarantee that it will work.