Thank you very much for your effort.
Unfortunately, the thing with search engines I did already - it does not help.
Proxy settings are "No Proxy".
The router is not the problem, because there are several smartphones that use the same router on everyday basis - and they all successfully connect to websites of interest (Google and Codeskulptor, for example).
This question sounds quite stupid, but since we are running out of options... is there any other, "secret" - "only root", way of setting network data? Could there be some little file somewhere telling my browser to go somewhere else?