A stock (or a fund) has an average return of 0%. It moves 
   on average 1% a day in absolute value; the average up move 
   is 1% and the average down move is 1%.

How does that yield an average return of 0%?

The usual way to do this is to take the natural log, so an up 1% day followed by a down 1% day (or vice versa) will always net out to a 0% change.

It goes up a little more often than it goes down?

Yes, start by writing a confusing question. One that starts talking about average up moves and average down moves and then switches to asking about the standard deviation of moves. Then publish a paper showing that people were confused by your question. Now you have "research" to back up your claim that everyone is confused by "mean deviation" and "standard deviation".

Thanks! Always nice to read the full research.