Thanks for trying everything & all the excellent feedback! The screenshots were very helpful, too. I noticed one of them said Download Failed, so I think the problem is probably more with Origin & downloading than with SimCity.
A couple of important things - be sure you are not running out of space on your C drive. And make sure your time/date is set to the region your computer is set in (Hungary, I think?) If possible, reboot your router/modem.
On your computer, close all your programs including exiting Origin (from the Origin menu, don't just click the X). Then clear Origin's cache following the instructions here: https://help.ea.com/en-us/help/faq/clear-cache-to-fix-problems-with-your-games/
Next, flush DNS cache as described here: https://help.ea.com/en/help/pc/connection-troubleshooting-advanced-pc/
Then cross your fingers, log into Origin again (as Administrator) & right-click the SimCity icon. If there is a "Repair" option, choose that. If not go ahead and click install.
I'm sorry you're having so many problems. Please let me know the results. 🤞
EDIT -