Totally unresponsive Google Play Services

Hi all. I've got a Samsung Galaxy S4 GT-I9505 on Lollipop 5.0.1. My phone has refused to connect to Google Play Services. I tried to add a Google account to my phone but couldn't. That was when I first noticed the issue. I cleared cache and even data of Google Play Services (and Google Services Framework) to no avail. I then went as far as to disable Google Play Services on my phone and then re-enabled it (according to the play store on the Web I've got the latest version). Still not working. And now I don't even have any google amount on the phone. Apps like Google Now and the Play Store don't don't even really open again.

Everything works fine on the Web (even using Chrome). When I try to add an account now, it recognises my email address and password (I deliberately tried a wrong combination and it was detected immediately) but gets stuck at the end of the process (pls see pic attached). What do I do?

On a separate note, I tried to do a factory reset but the phone just keeps displaying "shutting down" without actually shutting down (although I find some things like the phone time do get reseted). What do I do?

Finally, can't use Smart Lock anymore. The Smart Lock screen displays just the Smart Lock title up top. The screen proper remains blank. What do I do?

But by far my major problem is the Google account issue. Thanks.