Internet Accounts - Login Issue with Google and iCloud Google account issue

I cannot login with my Google account on El Capitan (OS X 10.11.1). It accepts my mail, password and 2FA verification code too (Google Authenticator). 

After pressing the last Enter, the login window turns grey and nothing happens (no feedback, no spinning wheel etc.)

I've left it for an hour without doing anything else, but still the same. 

iCloud issue

The other account I am having issues with is iCloud. iCloud is signed in (my Clear to-do list which uses iCloud for sync is syncing as usual.

However, when I go to System Preferences > Internet Accounts > iCloud, opens for a few seconds than locks itself. 

After entering my password it unlocks for a second, than "locks" again (the checkboxes become gray and unchangeable). 

I'm also unable to change AirDrop to Contacts Only from Everybody. Previously AirDrop had the No One status and when I tried to change it asked for my iCloud password, but didn't change after entering.

Facebook and Twitter accounts are both logged in with no problem on my Mac. Furthermore, I can access both my Google and iCloud account through web browser.
I'm certainly no expert on iCloud, but I recently had some luck by using the code (as opposed to the password) from my mobile device. You might try that. While it may not fix it, it may give you a more helpful error. That is what happened for me.