A face-down creature has no name, so it doesn't have the same name as anything else.

Although token creatures won't get a bonus from Mirror Box's last ability, tokens that are copies of nontoken creatures improve the bonus those creatures get.

The “legend rule” is the rule that states that if a player controls two or more legendary permanents with the same name, that player chooses one of them, and the rest are put into their owners' graveyards.

While the “legend rule” doesn't apply to permanents you control, you can control any number of legendary permanents with the same name and none of them will be put into the graveyard.

If you control more than one legendary permanent with the same name and the “legend rule” begins applying again (perhaps because Mirror Box leaves the battlefield), you'll immediately have to comply with the rule and put all but one of those permanents into the graveyard.