@woozle YES! I have that setting selected.

Here caption.

What option should I choose instead? Static IP?

  @ihshinron 

"Static IP" won't work for a typical consumer Internet connection. 
Since you obviously had Internet without selecting "IPv6 PPPoE", we can assume that is not what your ISP requires.
"6to4 Tunnel" would be used if your ISP didn't support native IPv6.
That only leaves "Dynamic IP(SLAAC/DHCPv6)", which is what I use, for example.

But better than guessing the correct IPv6 settings would be to follow instructions provided directly by the ISP.
 

  @woozle You beauty! It all seems to work now. I've selected the Dynamic IP(SLAAC/DHCPv6) option. Saved and then hitted the Renew button. Waited a minute or so and everything started to work as it should. THANK YOU!

Internet seems to be working fine on other devices.

Could I expect some older devices to have some issue with wifi connectivity or internet access after the change done?

Thanks again!

  @ihshinron 

The change shouldn't affect devices because of their age.

However, in case you notice further things not working as expected, then you might try with IPv6 switched off completely.

For example, we know that on most current TP-Link routers the "Speed Limit" feature only works for IPv4 traffic, but does not limit the speed of IPv6 traffic.