Hall of Echoes

The "legend rule" doesn't use the stack and can't be responded to. Notably, this means that if you have two legendary permanents with the same name on the battlefield before you have resolved the last ability of Hall of Echoes, there is no time to activate the ability before having to put one in the graveyard.

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. Notably, this is a state-based action that's checked whenever a player would get priority. It doesn't happen during the resolution of a spell or ability. If Hall of Echoes becomes a copy of a legendary creature, both permanents will remain on the battlefield.

Because Hall of Echoes isn't entering the battlefield when it becomes a copy of the creature, any "When [this creature] enters" or "[This creature] enters with" abilities of the copied creature won't apply.

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, you'll immediately have to comply with the rule and put all but one of those permanents into the graveyard. If this happens due to the effect expiring at end of turn, this will happen during the cleanup step, then each player will get priority and a new cleanup step will happen.

If the copied creature is copying something else, then Hall of Echoes becomes a copy of whatever that creature copied.

Hall of Echoes copies exactly what was printed on the original creature and nothing else (unless that permanent is copying something else; see below). It doesn't copy whether that creature is tapped or untapped, whether it has any counters on it or Auras and Equipment attached to it, and so on.

Both effects of Hall of Echoes's last ability have the same duration, and they'll wear off at exactly the same time. As such, there's no moment at which Hall of Echoes is still a copy of the target creature after the "legend rule" begins to apply again.