There are 2 definitions of "round" in 5e. There is the definition of "round" that you quote, which is from initiative position intMax to initiative position 0 (or negative if you manage that some how).

Then there is the definition used here for readied actions. This is the same definition used in "once per round" effects such as certain powers. This definition begins at the beginning of your turn and ends at the beginning of your next turn.

Because (unlike in 4e), readying an action and activating it, doesn't move your initiative order position, and because it matches the readied action refresh timer (which happens to use the same 1/round definition of round), readied actions can be used at any point in either the current round or the next round prior to your turn.

Here's how reactions refresh:

When you take a reaction, you can’t take another one until the start of your next turn. If the reaction interrupts another creature’s turn, that creature can continue its turn right after the reaction. (Player's Basic p70)

This is a large part of the support that leads me (and many others) to believe that readied actions can roll up to your next turn.

Absolutely

Once initiative has been rolled and your team is in combat, there's no longer a possibility of surprise. At that point, those hidden enemies likely are following their own initiative rolls, and in a sense, are participating in the same combat you are, if only in the sense that you can't see them and they're choosing not to attack.

In 5e, it is assumed that you are paying attention to all angles around you, so as soon as a new enemy shows up, you can definitely shoot at them with your held action.

Keep in mind that since you are focusing on attacking as quickly as possible, by spending your action to "be ready" for an ambush, your DM may not allow you to use a proper Perception check since your action is already expended, so you may not notice as soon as your party members do.

A standard trigger for a held action would be "If an enemy walks towards me, I attack it", which could easily convert to "If I notice a new enemy" or "If my allies point out a new threat".