So, I managed to fill in the contact form on dlink support site for my country, and spoke with a very nice gentleman (Adrian Apostol) who said to talk to the store where I bought the router from to get a RMA, since he saw the video and his opinion is that the unit is faulty. I contacted the store and told them about my problem and I am waiting for an answer.
In the mean time, I received a new firmware, 1.07 beta from mtztraktoros and told myself to try it, it couldn't hurt. Reset to default settings, upgrade to 1.07beta, reset again, configure the connection, and everything worked fine, even the PC plugged in the ethernet pory. I was very happy.
Then I thought that if the router is ok, it will work with any firmware, like before. Reset, install 1.05beta2 (the latest from the site), reset, configure internet, and the problem still exists. No other settings were made, beyond PPPoE user/pass and default wifi networks. I suppose if I go back to 1.07 it works, but I do not know for how long. Something is wrong with this router.