Baxi boiler pressure too high, fills itself?

The pressure on my Baxi boiler is too high and the release valve keeps opening. When I switch a tap on (cold or hot), the pressure goes down. When I switch the mains water off, the pressure goes altogether, and - even though the feeder tap is off - the system refills itself when I switch the mains water back on!

For a while, the system wasn't working at all (the radiators weren't getting hot), and it turned out the filter in the boiler was clogged with rust. After cleaning the filter, the radiators now work, but I'm getting rusty water from the taps.

Something ain't right, that's for sure, but what is it?