I made a move (black) which I can see was registered. The next thing I know my opponent is thanking me for offering him a draw! I knew nothing about it; is it even possible to offer a draw when it isn't my move? Surely if I clicked the Offer Draw button by mistake it would ask me to confirm?

@John-Pollard
sometimes when you do a "conditional " move, there is a box to check that allows you to accept a draw, if your opponent offers it. That might have happened to you.