Do a test. Download the Opera installer, launch it, click "options", set "install path" to a folder named "Opera Test" on your desktop, set "install for" to "standalone installation", uncheck "import data from default browser", and install. Test in that Opera, but do not enable any extensions, do not enable Opera VPN, do not enable Opera's adblocker and tracking protection, and do not enable Opera Sync?
Do things work fine there at least? If not, I'd look for something outside Opera causing the problem (time, date, time zone and daylight savings setting issues, malware, anti-virus-firewall). Of course, it it happens on mobile too and you're not using Opera Sync, it might just be an issue with your Google account itself, even if you can log in with Firefox.
