A: Utilizing the Mystery Play Free Action is not one of your regular turn plays, the first of which must be random. In fact, you can take your Mystery Play at any time during your turn, any time from before you even draw your first card, to after your last play (but before the next person goes).

Check out this chart of the order of events in a Fluxx turn (also linked at the bottom of the right sidebar) and understand that the Mystery Play, as a Free Action, can happen at any time in that area bracketed in purple.