Can you do Pidgeot EX's "Mirror Move" attack if the opponent's last attack "placed damage counters" (as opposed to "doing damage")?
No. Placing damage counters is an effect. Mirror Move does not copy any other types of effects from the last attack either.
Source: Evolutions FAQ (2016-11-03), TPCi Rules Team (2016-11-03)

Relevant cards

Note: All applicable cards may not be listed below.
XY12_104
XY12_64