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If the point of quenching were to cool it as fast as humanly possible, we'd use liquid nitrogen. Quenching is the process of cooling it at the appropriate speed — the faster you quench, the sharper the blade and the more brittle the blade. The slower you quench, the softer the blade but the less likely it is to break. That's why katanas have a hamon — the edge was quenched in water quickly while the back of the blade is coated in clay and cools slower. This gives the katana a relatively sharp edge with a more flexible, stronger backing without using significant extra steel (iron/steel was rare in Japan, and had to be conserved, as opposed to Europe where the backs of single-edged blades were just thicker for the same reasons).

So, the goal with quenching is to get the right balance of ability to hold an edge (fast cooling) and ability to take a blow without shattering (slow cooling). Mercury is likely to cause too many headaches to provide any significant benefit.

Also, I'm not a metallurgist, but I know that most of the mercury-metal reactions I've heard of don't stay on the surface, but the mercury travels deeper into the metal (see: mercury-aluminum, mercury-gold, etc.) so polishing off any problems might not help.

The point of quenching is to use an heat sink to rapidly subtract heat from a hot piece of metal, so that a certain phase transition happens. When applied to steel, quenching is used to freeze an otherwise unstable crystallographic phase, and the freezing is due to the rapid cooling.

The boiling point of mercury is, as you note, higher than water. This implies that the freezing would be slower or not happen at all. A slower freezing might be wanted for metallurgic reasons (lower stress on the structure, better properties for the particular usage) and this is the reason why sometimes specific oils are used, but no quenching at all simply defies the purpose of quenching.

About the formation of amalgama, I think you are underestimating the extent of the damage: the quenched layer is the superficial one and the effect of quenching fades away while going more in depth, thus scraping away the surface would remove also the hardened layer. Plus, removing a layer from a stressed material will likely result in induced cracks. For this very reason quenching of worked metal is done after all material removing steps in the production process.

A couple of notes: mercury generally doesn't form amalgams with iron, see for example https://chemistry.stackexchange.com/q/28916/54697 where a reference is given for low-temperature amalgams. My recollection is that it was sold in units of steel 76-lb flasks so one might think that the mercury isn't going to mess up the steel very much. Now if meteoroic iron were used with its relatively high nickel content the situation might be different; I don't know.

Water might not be as effective a coolant as one might think due to the Leidenfrost effect. I know that I have stuck my fingers into liquid nitrogen for a couple of seconds without harmful effects, but doing the same with liquid propane may be more risky if not immediately injurious. Lower boiling temperature doesn't necessarily imply faster cooling. Don't try either experiment!

Looking at Bergman, Lavine, Incropera, and Dewitt, Fundamentals of Heat and Mass Transfer, sixth edition, John Wiley and Sons, New York 2011, p. 581, we see that
$$overlineNu_D=fracbar hDk$$
Where $overlineNu_D$ is the average Nusselt number for a cylinder of diameter $D$, $bar h$ is the average heat transfer coefficient, and $k$ is the thermal conductivity of the fluid. Then a correlation for $overlineNu_D$ is given as
$$overlineNu_D=left0.60+frac0.387Ra_D^1/6left[1+
left(0.559/Prright)^9/16right]^8/27right^2$$
For example, at $400,°C$ table A.5 gives a Prandtl number of $Pr=152$ for engine oil and $Pr=0.0163$ for mercury. The Rayleigh number is given on p. 573 as
$$Ra_D=fracgbetaleft(T_s-T_inftyright)D^3nualpha$$
Where the gravitational acceleration $g=9.81,m/s^2$ on earth, the volume thermal expansion coefficient for oil is $beta=0.70times10^-3/K$ while for mercury it's $beta=0.181times10^-3/K$, and maybe the difference between the surface temperature and the free stream temperature is about $T_s-T_infty=500,K$. We are hoping that the sword will be reasonably approximated by a cylinder of diameter $D=5,cm$ for the purposes of convective heat flow analysis, the kinematic viscosity of oil is $nu=10.6times10^-6m^2/s$ and for mercury $nu=0.0882times10^-6m^2/s$ and the thermal diffusivity of oil is $alpha=0.695times10^-7m^2/s$ and of mercury is $alpha=54.05times10^-7m^2/s$.

Plugging in all these numbers for oil we get $Ra_D=5.82times10^8$ so $overlineNu_D=136$ while for mercury $Ra_D=2.32times10^8$ and $overlineNu_D=32.8$. But it's the average heat transfer coefficient that is proportional to the rate of heat flow and for oil the thermal conductivity $k=134times10^-3frac Wmcdot K$ so $bar h=364fracWm^2K$ while for mercury $k=9800times10^-3frac Wmcdot K$ so $bar h=6430frac Wm^2K$.

Metals can use their conduction band electrons to conduct heat as well as electricity so mercury looks like it might be $20times$ as effective as oil at transferring heat by convection. Now, whether this is a good thing or a bad thing is a question for the blacksmith to answer.

Not for the blades.

Water has a very good feature - it boils at 100 C, quickly taking away a large amount of heat which would not be possible with liquids that are boiling at a higher point.

Quenching in a high point boiling liquid like oil is definitely a thing, but this leads to lower hardness - not something we would want for a sword blade.

However, mercury can definitely be used in other metallurgical processes directly related to quenching - soaking and tempering, which require temperatures higher than 100 C.

In addition to the density and specific heat issues raised in other answers, steel is susceptible to liquid metal embrittlement. If exposed to a liquid metal (say, mercury) and at the same time a tensile stress (say, from quenching) catastrophic fracture will occur.

If you are interested in a steel treatment that is toxic to workers, look into cyanide salts. They are both a great way to carburize steel and extremely toxic. (Carburizing adds carbon just to the surface, creating a part with a harder wear-resistant surface while maintaining a softer crack-resistant core.)

No

Mercury offers no advantage over oils and has many moe drawbacks. the vapor point of mercury is lower than many quenching oils, so sou can't get it any hotter than oil. Cooling faster offer no benefit if you are still cooling to the same temprature, in fact it makes it worse. It leads to stress fractures, even water cools too fast for better quality steels, and waters thermal conductivity is nearly an order of magnitude less.

The toxicity cannot be hand waved either quenching is a technical process it can't be done by unskilled labor and it will kill your skilled labor very quickly. There is no medieval protection from mercury vapor.

Lastly getting enough mercury to actually quench a blade would cost more than a small army, mercury ores contain very small amounts of mercury, and are not easy to process.

I have made several swords and repaired them many times. I have tried several methods of quenching but nothing beats just throwing it in a fast flowing stream. The only comparable sword was a rough clip sword I hastily made out of scrap mild steel from a building site using a mold carved in a block of slate and quenched in the sea. I usually case harden my swords in a mix of charcoal, hay and animal bone. The finished result was as good as any EN45 sword I have.