but in the past you needed to log in to your phone at least once a month for the web session to keep working. Is it that still the case?

Yes, unfortunately. From my limited experience trying to get it off a smartphone, leaving the number on a dumb phone and trying to activate on a sim-less smartphone, it simply won't work if doesn't detect a mobile number in its host Android

It doesn't use servers to keep messages, so when you first access from outside the "original device", it downloads the last few received messages of every conversation still in your phone, but rarely everything.