The population estimate I have read for the Carpathian basin - The Kingdom of Hungary - was 3 million before the mongol invasion. That lives a lot of empty space - many of which was forested at the time. There were also swamps along some of the rivers and extensivly in the southern Banat.
About the royal ring: its rather hard to imagine that when the mongol have especially tried to capture the king they wouldnt have the wits to use his ring if captured. They have destroyed the army of the king and were able to do as they pleased without serious opposition. Its pretty easy to imagine that they found someone who was willing to help them with the letters. And they dont have to inform every village themselfs - just the counties. They will be the one to inform the villages so the local administration does that for them.
And even if we go with the most modest estimate - with the only reason for doing so being your weak imagination - we are speaking of 15-20% of population loss. 2 man in every 10 dead. Thats already plenty brutal if you ask me. Also if sources agree about the mongol brutality from Central Asia through Russia to Hungary maybe we might believe them.
Usefulness of the swamps as the hiding places for the people with their livestock and food supplies is limited (what your cattle is suppossed to eat and where are you going to stay?) and, as you noticed, they were not conveniently located all over the country so most of the population in the raider area simply could not use them. The same goes for the forests: Central Russia was heavily forested but the Mongols had been operating there without major problems (so-called “forest Mongols” lived on the fringes of taiga, which is more than a match for any European forest). Anyway, you can always find somebody showing you the path to the hiding places. If there are willing cooperators ready to write the letters (without cheating with the content), to carry these letters to the regional centers, deliver them and report about the false victory (with a high risk of being contradicted by fleeing survivors and tortured), why do you think that the lower classes would be less prone to a cooperation if promised a personal security? What would be the reason for the Mongols to kill most of the local population on the territories they hold (which would be only a part of Hungary)? Just for the fun of it even if this meant that these dead people would not be producing food for the conquerors?
Now, as far as the estimates are involved, the contemporary reports, from China to Hungary, are known for the gross exaggerations all the way to a complete annihilation of a local population being reported by a (presumably) local eyewitness. Then we have to separate the losses from the invasion from those of the unrelated causes: very cold and wet winter with a flooding of a big part of the Hungarian Plain and resulting famine (the Mongols hardly could be blamed for the climate). How many people died from each of these reasons? Do you have any meaningful breakdown or just repeating the numbers without thinking? How many of the reported losses were the people taken prisoners and sent Eastward? The Mongols tended to do this to the artisans. I understand that these people are still “losses” but they were not killed as your post implies.
Now, was 15 - 20% something unheard of during the MA? Judging by an endless list of the cities destroyed in the Central Russia, percentages there should be at least on the same level (and, contrary to all evidence, practically all princes were reported killed) and if we believe the more or less contemporary estimates for the CA, the losses here would probably exceed 100% of the reported population (just for the case of Bukhara, one has to believe that it had over a million of population and was completely depopulated after being taken by the Mongols). If you believe these stories, I can sell you the Brooklyn Bridge really cheap. But let’s accept the number, just for the sake of the arguing. Russian principalities had been back in business and even easily taxable immediately after the Western Campaign (in other words, as soon as Batu had time to establish at least some administration). The same goes for most of the CA.
So why the same or lower level of the losses and destruction would be catastrophic for Hungary where the Mongols stayed only for few months and which they did not tax afterwards?