Hi Michal
I don't think that wireless bridging is what you need to be doing. Correct me if I'm wrong: with your PC you can connect to your Sweet wireless router - you can see the router's configuration pages via your web browser (Internet Explorer probably), but you cannot see anything on the internet. This means that the wireless part of your network is functioning properly and that DHCP is also working over your wireless network i.e. your router is allocating an IP address to your PC.
So this leaves the connection between your cable modem and your router. Forgive me for going back to basics but check that the cable from the modem is plugged into the WAN port on your router. If that is all ok then check the WAN Setup on your router config page: it should be set to "Dynamic IP User". This allows the router to get an IP address from your cable modem. Once this is all set correctly try powering-off and on JUST the cable modem.
Good luck!