How many different MOV files have you tried playing? I wonder if there might be an issue with the particular MOV file you've been trying to play. Also, MX Player just got an update, and although they don't specifically mention this issue, it's worth trying again after updating it.
Your entire Picasa photo collection is probably just being accessed in the cloud. If you pick a photo to look at, I think it's just temporarily caching it in your system, unless you specifically download it.