Hi Capetonian
The answer to all your questions is no. I'm typing this on the laptop right now, so obviously it will access some internet sites, but not the ones I really need.
The scan with AVG was done with the up-to-date version as well.
The sites themselves are not the problem, as I can access them through iphone and android tablet, but, (especially with the banking), full accessibility of the site has to be done through visiting the full website on a computer.