What is the ruling on downloading a video or entering a website that displays a cross or flags of countries containing a cross, knowing that the device saves images of the cross or the flag containing a cross when the site is loaded?

It is permissible to benefit from and download from the website if you need to, and if you can remove the cross from your device, then do so. The scholars have stated that it is disliked to have crosses and that they should be removed from clothes; this is based on the hadith (definition) of Aisha (may Allah be pleased with her) who said: "The Prophet, peace and blessings of Allah be upon him, would not leave anything in his house that had crosses on it without destroying it." It is also possible that placing a cross image on clothes or similar items is forbidden.