As already commented by Andy: You are sampling at 500 Hz. The mains frequency is close to, but not quite, 50 Hz. Importantly, your sampling rate is not synchronous with the grid frequency. This means you will have a very strong aliased tone very close to 0 Hz. It is this tone that you are observing in the scope.
All of your observations can be explained, if you think about how the waveforms look in each case, and which section of it is most prone to be undersampled.