There are two very disparate answers here to be done. To the first, it is a fairly straight forward one though. The duel of honor has always been illegal, and that is in fact a defining feature of it, the concept taking form in the Early Modern Period as a response to the rising refusal of rulers to 'grant the field' for a sanctioned fight.

Now, while in practice the duel has rarely been prosecuted, and when prosecuted, the duelists rarely have faced serious legal penalty, there was at the very least an understood necessity to not be open and blatant in the planning of the duel, for the most part. Plans for the duel ought to be kept secret, especially the time and the place, as if they caught wind of it, authorities could act to stop it from happening. Depending on the specifics, this might be through the arrival of law enforcement prior to the appointed time, or it might be through the courts, with the duelists bound over with a bond of some monetary amount to hold the peace.

Dueling at the crack of dawn was, most simply one of the methods used to at least avoid the former. It minimized the likelihood of running into anyone else who might interfere with the task at hand, and for that matter could often give the authorities an excuse to not interfere if they were so inclined by simply arriving too late. There was nothing particularly symbolic in the choice of dawn by many duelists, it just was convenient!

Now, as for the 'duel at high noon', this is a bit more complicated, but not because there is some complicated symbolism in why cowboys would choose to face off at that time of day, but rather because they didn't. I've written previously in this thread about the mythos of the duel in the American West, but to recap it in brief, our stereotype of the shootout in the Old West, with two duelists facing off for a quick draw contest at high noon in the middle of town is basically made up. The American West was quite violent, but that is the creation of dime store novels, and there are probably some Western films out there which have more duels like than than there were in the entire 'Old West' period!

One of the few shootouts which comes close was the Hickok-Tutt duel, which was fought in Springfield, Missouri in 1865 and did see the two squaring up against each other downtown where they did draw and fire at each other, but the duel itself was not fought by some arrangement to meet at high noon. This fight was the prototypical 'quickdraw' showdown though, and absolutely can be seen as a critical underpinning of the entire trope to be found in the Western genre, but whereas the 'classic' duel fought at dawn is quite firmly grounded in reality, the meeting at high noon is much better understood as one more piece of literary flourish added onto a mythical view of the American West that never quite comported with reality.

Sources

I maintain an extensive bibliography on dueling which can be found here. For this topic in particular, I would draw attention to the works of Dykstra and Slotkin for the broader study of violence in the Old West, as well as Steward for more specifics on the duel (in Missouri). For the duel, Banks is particularly relevant here in its focus on dueling in England, as well as Kelly for a focus on Ireland.