Once Deepfathom Echo becomes a copy of another creature, it no longer has its triggered ability until end of turn.

Once an ability that causes a creature to explore begins to resolve, no player may take any other actions until it's done. Notably, opponents can't try to remove the exploring creature after you reveal a nonland card but before it receives a counter.

In some unusual cases, noncreature permanents may explore. For example, if the creature card returned by Defossilize is somehow not a creature once it's on the battlefield, it can still explore. You'll take all the same actions, and you may end up putting a +1/+1 counter on the permanent. (Note that some effects target a creature, and those effects would still require a legal target to have it explore.)

If an ability instructs a creature to explore, its controller reveals the top card of their library. If it's a land card, they'll put it into their hand. Otherwise, they'll put a +1/+1 counter on that creature, then choose to either leave that card on top of their library or put it into their graveyard.

If a resolving spell or ability instructs a specific creature to explore but that creature has left the battlefield, the creature still explores. If you reveal a nonland card this way, you won't put a +1/+1 counter on anything, but you may put the revealed card into your graveyard. Effects that trigger "whenever a creature explores" trigger as appropriate.

Some spells or abilities might cause a creature to explore multiple times in a row. If you reveal a nonland card when a creature explores and leave it on top of your library, then the creature explores again immediately afterwards, you'll reveal the same card again.

If the copied creature has in its mana cost, X is 0.

If no card is revealed, most likely because that player's library is empty, the exploring creature receives a +1/+1 counter.

Non-copy effects that have already applied to Deepfathom Echo will continue to apply to it. For example, +1/+1 counters on it will still affect it.

Deepfathom Echo copies exactly what is printed on the chosen creature (unless that creature is copying something else or is a token; see below). It doesn't copy whether that creature is tapped or untapped, whether it has any counters on it or any Auras and Equipment attached to it, or any non-copy effects that have changed its power, toughness, types, color, or so on.

If the copied creature is a token, Deepfathom Echo copies the original characteristics of that token as stated by the effect that created that token. Deepfathom Echo doesn't become a token (although it might already be one).

If Deepfathom Echo becomes a copy of a creature that is copying something else, it becomes a copy of whatever that creature is a copy of.