GSP, to be perfectly honest, this is a really, really difficult question. I really don't think I know enough of the chemistry of urea to give you complete satisfaction. I can take a crack at it, but it's on a best efforts/commonsense basis:
The fizzing is from the urea being oxidized to carbon dioxide and the nitrogen oxides going to nitrogen and water, with formation of two equivalents of ammonium cation. Also recall that urea is a base (see those two amines on the right and the left, both of them can be protonated). What steps really happen when you put the urea into a strong, strong
oxidizing mixed acid solution would be guessing on my part, the obvious is that it breaks up in the presence of H+ and NO2/NOCl. As for what happens first, that would require some thought on my part. I believe that it is necessary that it be in very acid conditions (as urea is a relatively stable molecule all things considered) to be taken apart. Clearly, chlorine plays a role in this chemistry as it is what is responsible for ensuring that most of the nitrate from the nitric acid is in a form that is active and ready to be reduced.
I'm waiting on Noxx to put in a program called LaTeX (it's an in-board equation editor). After that, I will take a stab at the mechanism and the equations for it. I can say this for sure, it is acid-base first with protonation occurring before oxidation (this almost always occurs first). This is all driven by the main fact that gas is produced, and secondarily, ammonium cation is formed. Curiously enough, if there is enough residual NOCl present, ammonium is oxidized into nitrate (well nitric acid probably).
I've only used urea on a few occasions and I didn't like it. Technically speaking, it should be absolutely clean and never leave residues but like you said, it's easy to over add and that is what causes problems (especially when PGMs are present).
As far as the standard ways go, adding urea until the pH is raised to 1 seems rather more scientific than until it just stops gas production since they probably have the same underlying chemical reason. For me, that implies several possibilities: 1. at pH 1, the solution is no longer acidic enough to properly ensure urea's decomposition to gas, 2. at pH 1, the solution is no longer acidic enough to sustain significant concentrations of NOCl which further implies that there would be small/negligible amounts of active chlorine.
When you put nitric and hydrochloric acids together (in any ratio), the end product is a reduction-oxidation reaction with nitrate oxidizing the chloride from HCl to chlorine, again this only occurring in highly acidic milieu. Lots of things come out of an aqua regia solution: chlorine gas, nitrogen oxides, and even oxygen (which usually oxidizes any NO).
2 HCl + 2 HNO3 --> 2 NOCl + 2 H2O + O2 <--> 2 NO + Cl2 + O2 (+ H2O) -->2 NO2 etc
It is not a simple situation because even above, things are in equilibrium, and there are competing reactions.