Update on this issue, I tried a couple of other things that I found could help.
- I uninstalled both the .net framework windows features, rebooted, and reinstalled/updated them to see if they might have been corrupted.
- I reinstalled Adobe Air as I read somewhere that TS3 uses it.
- I updated windows 10 to the newest version (it was available so why not)
- I uninstalled and reinstalled my WHOLE game, which took a LONG time, including trying if a fresh game with a fresh game folder would help
- I flushed the DNS Cache with Command Prompt run as Admin
And yet again, nothing. I talked to an EA representative on the phone who sent me an email with instructions among which it said to flush the DNS Cache by enabling the admin account. However, I accidentally removed the wrong email among the many that I received from my tickets, so I looked up flushing the DNS. There I found instructions on doing it by running CMD as administrator. It said it was flushed, I am not 100% sure if that is a big difference or not.
I think / fear that I tried everything apart from reinstalling windows, which I think should never be necessary to fix something as 'small' as not being able to log in to a game server. So if anyone still has ideas as to what else I could try please come forward. I just don't want to believe that this issue can't be solved with one thing or the other!