Three allies and I are in a room and an enemy just went underground. Can I ready my action with my bow saying: "When he pops out from underground I attack"?
What happens if he pops up behind me? Do I still get the attack?
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If your trigger is "the enemy pops out from underground" and your character can perceive it, the trigger occurs and you can take your readied action.
Facing is an optional rule in the DMG and as long as your table does not use it, there is no "behind you".
Seeing someone pop out from underground is not the only way to perceive it, as long as he isn't mechanically hidden and your character isn't mechanically deaf, you can also hear it. Additionally even if he is hidden, as soon as he comes into plain sight (neither heavily nor lightly obscured) he instantaneously gives away his position and is no longer hidden.
Yes, you can cover a whole room with your trigger, as long as it's perceivable, but keep in mind that moving is not part of a readied Attack and you have to make sure the requirements of your Action are also fulfilled. If they are not, the action is wasted.
That means for your attack he has to be in range and not behind full cover.