Yes, not to do so would be the worst possible conduct to engage in, as the very underpinnings of the duel were about demonstrating ones manhood through the courage of putting your life on the line. For the most part, the ground rules of the duel were intended to prevent this from happening to too great a degree, with firing usually regulated to occur within a right window, meaning even if one duelist fired first, the other would have, at most, 1 or 2 seconds in which to shoot or else lose their fire, but this wasn't always the case. The convention took time to develop, and didn't take hold everywhere, such as in Germany, where it was expected duelists would take their time, and could spend up to a minute taking aim before firing.

In the Anglo-American tradition though, where taking your time was at least frowned upon, we do have examples of duels without a count off where essentially what you envision happened, perhaps most famously the duel between Andrew Jackson and Dickinson, where Jackson used it to his advantage by letting his opponent shoot first - and quickly - so that he could take his time in lining up the shot. Accounts do note that Dickinson tried to back up, but was ordered to return to his mark by the Second, something which was the duty of the Second, who, in theory, would be justified in shooting down a Primary for an egregious violation of the code. To be sure, taking slow careful aim was considered poor form, but without an agreed upon count off, technically not in violation.

For further expansion, this older answer of mine goes into more detail on the mechanics of regulating the duel and the attempt at equality, so should be of interest, as well as this which looks at the related phenomenon of reserving fire and deloping. This one looks at the nature of the challenge and the underlying concept of honor inherant in the duel.

Effectively, yes! However, it is also important to understand that the intent of dueling wasn't necessary to kill or injure your opponent, but to stand and meet the challenger with the intent of defending your honor. One of the ways that you defended your honor was by being able to stand there unflinchingly while your opponent took aim and fired.

Pamphlets and short books were written on the subject and outlined the expectations for the duelers. To draw upon a copy of "The Art of Duelling" by "A Traveller" (published in London in 1836 and advertised as having "... much information useful to young continental tourists") the need to properly prepare in pressed upon the reader (p. 47):

If a pistol misses fire the party loses the shot: he cannot, under any circumstances, be permitted to fire again. It is therefore very requisite to be particularly careful in charging your pistol, and fixing the cap, that this mishap may not occur.

Firing early also requires your to stand your ground (p. 44-45):

Instances have not unfrequency occurred, where, through a misconception of the signal, one party has fired before the other; in these cases, the second party has undoubtedly a right to his fire.

Finally, the author has the following to say about deloping (i.e., firing into the air) (p. 48):

Upon a delope the affair immediately terminates and the Seconds should never permit another discharge. When a man fires in the air it is considered an acknowledgement that he has been in the fault; and although he may still refuse to make an apology, the opposite party has no right to demand another fire.

Author expands upon this by also warning the reader to keep any intents to delop quiet since it is easier to aim when you know the other person isn't aiming at you!

Reading between the lines of the "The Art of Duelling" it is possible to pick-up upon the idea that the intent of the duel wasn't to kill or injure your opponent, but rather to defend or recover honor following an insult (i.e., wounds of honor may only be cured with steel). The author notes that, based upon their reckoning that odds of a fatality were about 1:14 and injury was about 1:6 (p. 32-33). It was also recommended you stand with your right side facing your opponent and minimize amount of area that you present as a target (p. 25-27). Likewise, the recommended time of day, design of pistols, distances, and methods of aiming are more likely to reduce the likelihood of being struck as opposed to increasing it. However, the author had the following admonishment in the event you were struck (p. 45-46):

I cannot impress upon an individual too strongly, the propriety of remaining perfectly calm and collected when hit: he must not allow himself to be alarmed or confused, but summoning up all his resolution, treat the matter coolly; and if he dies, go off with as good a grace as possible.