What is the ruling on downloading modified electronic games available for free from specialized websites, and is it obligatory to purchase them at high prices if it is impermissible to download them, knowing that these modifications may have been made without the consent of the producing company?

If you do not know that the owners of the programs available online permit their download, then you must research their permissibility. If you cannot ascertain, then you must avoid downloading them. As for electronic games, there is no harm in them if they are free from Shariah prohibitions and do not distract from an obligation, though excessive engagement with them is detrimental to one's honor and a waste of time. Whatever contains a Shariah prohibition is forbidden.