How does Painter's Servant interact with Mycosynth Lattice? I'm inclined to believe that colors would comply to the card that entered the battlefield most recently. If that's the case, what would happen if both were to enter the battlefield simultaneously?
It’s a matter of timestamp, i.e. which of the two is most recent, as you guessed. The newer of the two permanents wins.
As for multiple permanents entering the battlefield simultaneously, when that’s the case, the active player chooses the relative timestamp order for all permanents entering the battlefield together.