Yes, 0.3Ω is pretty much a direct connection.
I'm not sure what to suggest regarding checking your home earthing system. I guess, start by finding the earth stake and inspecting it visually. Then get a long piece of wire and attach one end to the stake, take the other end to the room with the mains socket, and check resistance from the earth pin on the mains socket to the wire. (You might want to check for voltage between them first, in case there's some high voltage that might damage the meter on ohms range.)
If the socket earth is properly earthed, stand on your carpet in bare feet, and measure AC voltage between your body and the mains socket earth pin.
Use your scope to look at the interference by just holding the probe end. Adjust the timebase and try to find out the frequencies of the signals it picks up. Some will be at 60 Hz I guess. There may be some at 120 Hz too. And it looks like there's something at a higher frequency. Try to identify the source(s). If there are any appliances nearby, try turning them off, unplugging them, or pulling the fuse or popping the circuit breaker for them.
Otherwise, get an electrician to do a proper check. Also you could try to find a forum used by electricians (rather than electronics people) and ask there.
Edit: You could also pop the cover off the mains socket (open the circuit breaker first) and check for spring contacts that are bent too wide so they don't make good firm contact with the pins, and/or dirty or corroded. Usually you can just clean them, sand them, and bend them back, but you might want to replace the socket.