If you take damage between the start of the upkeep step and the time in which the upkeep trigger resolves, then Soul Echo will not affect that damage and you will be affected by the damage as normal.

If this card runs out of counters before your next upkeep, the effect still replaces the damage and uselessly tries to remove counters from the now empty card.

If you have more than one Soul Echo, the opponent chooses which ones will be replacing damage that turn (if any). And when damage happens, you decide which one of the ones the opponent selected that you want the damage to be replaced with, but the damage all goes to one of them.

In a multi-player game, you choose a new opponent during each of your upkeeps.