1. It will enter the battlefield with four counters. Entering the battlefield with counters counts as having counters placed on it, so the Biomancer would cause the Mage to enter the battlefield with two counters, which the Menace will double to four counters.
2. Only two. The Menace has to be on the battlefield when the counters are being added in order for it to double those counters. Since it's entering the battlefield at the same time as the Mage, it won't double the counters that are placed on the Mage and itself.
3. The Mage does not get any counters. Just like in 2, the Biomancer has to be on the battlefield when the creature enters the battlefield in order to get the counters. Since all three creatures are entering the battlefield at the same time, the Biomancer won't apply to the Menance or the Mage entering the battlefield, so neither creature will get any counters.
Daniel's explanation linked to in the original post applies to these questions as well.
For those who don't know who I am, I'm Nathan Long, and I am also a NetRep (for the Wizards' forums). I will occasionally be answering questions here, and my answers should also be considered O, just like Daniel's and Scott's answers.
Edited Mark Brown (June 26, 2013 07:20:09 PM)