A cheap adapter will not alter the signal timing. HDMI/DVI are very much similar to VGA, having red, green, and blue signals (or YCC) plus a horizontal and a vertical sync signal. A cheap adapter is nothing more that a chip which receives the TDMS RGB signals and converts them to analog (possibly doing YCC->RGB conversion). The sync signals are passed through unaltered. This works fine when the signal is within the sync range of the analog (VGA) monitor you are using, and it fails otherwise. Horizontal resolution is essentially arbitrary for analog signals (resolution is simply related to bandwidth) but the vertical resolution is dictated by the ratio of the horizontal and vertical sync signals. I'd suspect that your monitor does not like the 50 Hz vertical refresh rate, or possibly doesn't like the vertical resolution or the size of the "porch" (blank) areas in the 1080p signal, as they are not quite the same as computers typically used.
You could try a smart converter which will capture the HDMI input into a framebuffer then scale that and output it at a resolution and refresh rate that you can choose. These are much more complicated and more expensive than the cheap ones which are basically just a few DACs.