Do Gild's Gold tokens go into play under the caster's control?
Yes, the token comes into play under the control of the player who played Gild. This is specified by rule 110.2a:


  If an effect instructs a player to put an object onto the battlefield, that object enters the battlefield under that player's control unless the effect states otherwise.


Gild's effect instructs you (the player who controls the spell) to put the token into play, so it comes into play under your control.

A ruling on the Gatherer page states this explicitly:


  You will control the token, no matter who controls the target creature.