No need to test as it definitely does. There is no known card in the game that allows a monster or spell/trap keep Counters of any kind once it leaves the field for any reason.
If I use the 2nd effect of ennemy controller (take the control of 1 oppononent monster for 1 turn)
and then banish this monster with this card, he will come back on my side of the field for all the game?????
Someone just used michizure after I destroyed their monster, and the game never asked if I wanted to use Interdimensional Matter Transporter in response to save my monster. Is there a different time for when I'm supposed to use it?
If they used Michizure during the damage step then you cannot use IMT during that time. Remember that only counter traps can be used during the damage step unless they specifically state otherwise (like Michizure)