Complete nonsense, I'm afraid.

The existence of the Mongol Empire and its successor khanates cannot seriously be questioned. To address the author's points in order (briefly, as I'm not an expert in this area):

1. The Secret History of the Mongols, dated from 1240, is the defining contemporary source on the life of Genghis Khan and the politics/culture/history of the Great Khanate.
2. It shouldn't be a surprise that an originally nomadic society (especially one whose empire was a cosmopolitan conglomeration of a massive range of cultures and societies) doesn't leave a significant architectural footprint. But there's always Karakorum.
3. I don't know anything about Russian linguistics, so can't comment on this one.
4. Russian culture is a unique fusion of east and west; Slavic and Turkic (and more besides.) The problem with melting pots is that deconstructing and taxonomising cultural influences is hard.
5. You can't talk about the Mongol Empire as if it just disappeared; it's not (primarily) an archaelogical question in that sense. The empire splintered into four successor empires — the Yuan dynasty in China, the Ilkhanate in Persia, the Golden Horde in the Urals, Siberia and parts of Eastern Europe, and the Chagatai khanate in Mongolia and China. They, in turn, rose and fell and dispersed — and with them, their assets.
6. There do appear to be Mongol coins.
7. See Mongol bows; an evolution from recurve/composite bows.
8. I don't know enough about Mongolian folklore to offer a detailed answer.
9. That's just not true. This is from the precis of a study by geneticists at the University of Oxford on genetic admixture:

How do your results on the Mongol expansion relate to previous analyses?

More recently, a study using genome-wide data with different methods and genetic markers but on a similar (but smaller) set of populations to those used in our paper (Patterson et al. 2012), found evidence of admixture in the Uyghurs, dating to the time of Genghis Khan. As well as the Uyghurs, we found evidence of this Mongolian expansion in a further 6 populations, all with similar dates, and sometimes much further west. These populations approximately span the maximum spread of the Mongol empire. There are many central Asian and Eurasian populations in our analysis that don't show evidence of Mongolian admixture, implying that most Asian populations were not affected by this expansion. Taken together, we believe that there is now strong evidence that this event had a major impact on many Eurasian populations.

The person writing this is obviously a Russian nationalist. I have met more than a few people who want to deny that Russia was ever under the control of the Mongols. It seems like it hurts their pride or something.

Most of the points have already been addressed very well but I will add a few things:

• 1. Mongolian adopted a written system from Uighur in approximately 1204. This written system is still used in China (in Inner Mongolia) and was in turn adopted by other languages (Manchu and Oirat, unfortunately now basically dead). This is incontrovertible fact. That said, the Soviet Union did introduce a system based on Cyrillic to the Mongolian People's Republic in 1946 which is the system now used there (however note <50% of Mongolians live in modern Mongolia). Obviously the author is either confused and does not realize that the Soviet system replaced an old one or he is just a Russian ultranationalist.
• 3. This is incorrect. There are many words in Russian from Mongolian and Turkic (the language spoken by much of the Mongol army) -

Language

While the linguistic effects may seem at first trivial, such impacts on language help us to determine and understand to what extent one empire had on another people or group of people – in terms of administration, military, trade – as well as to what geographical extent the impact included. Indeed, the linguistic and even socio-linguistic impacts were great, as the Russians borrowed thousands of words, phrases, other significant linguistic features from the Mongol and the Turkic languages that were united under the Mongol Empire (Dmytryshyn, 123). Listed below are a few examples of some that are still in use. All came from various parts of the Horde.

амбар                          ambar                                    barn
базар                          bazar                                     bazaar
деньги                        den’gi                                     money
лошадь                       loshad‘                                    horse
сундук                        sunduk                                   truck, chest
таможня                      tamozhnya                              customs

One highly important colloquial feature of the Russian language of Turkic origin is the use of the word давай which expresses the idea of ‘Let’s…’ or ‘Come on, let’s...’ (Figes, 370-1). Listed below are a few common examples still found commonly in Russian.

Давай чай попьем.              Davai chai popem.             ‘Let’s drink some tea.’
Давай выпьем!                    Davai vypem!                   ‘Come on, let’s get drunk!’
Давай пойдём!                    Davai poidyom!                 ‘Come on, let’s go!’

In addition, there are dozens of place names of Tatar/Turkic origin in southern Russia and the lands of the Volga River that stand out on maps of these areas. City names such as Penza, Alatyr, and Kazan’ and names of regions such as Chuvashia and Bashkortostan are examples.

Source. I don't recognize most of these words from modern Mongolian so I assume they are mainly Turkic.

• 6. There are Mongol coins. In fact there are Mongol coins in the excellent Moscow history museum if you want to see them along with a number of other Mongol artifacts.
• 8. Of course Mongolians have mentions of their great past in their folklore. There are countless stories about Chinggis, his generals, his wives, and his successors.

I think if you'd read the travels of Marco Polo, you'd be astounded to what degree of order you'd find in the Mongolian Empire under Kublai Khan. Amongst other things, you'd find that their postal system was incredibly advanced, with stations on every main road leading from Kambalik. Theese stations were set up every 25-30 latin miles (1 of which is about 1 1/2 KM). Theese were called "Jamb" and each one of them were large furnished houses dressed with silk tapestry etc, so that even exceptionally wealthy or highly regarded travellers could stay in them. At each of these stations, there'd be about four hundred horses ready at a moment's notice to bring a messenger or other curriers on their way with a fresh horse.

Marco Polo states that "Thanks to this, royal messengers could travel both easily and comfortably throughout every province in the empire, and all this shows that the Khan has greater power than all other emperors and kings and other people combined. His postal service employs no less than two hundred thousand horses and ten thousand buildings filled with all the needed equipment. The system is actually so wonderful and so perfect as you could only imagine"

I've translated this from a Norwegian copy, so excuse the somewhat strange wording here and there, the translation is from the early sixties which makes it kind of strange even in Norwegian. Reading Marco Polo's stories from when he was employed by the Khan leaves no question to wether or not the Mongol Empire was truly an empire, in fact it might well have been one of the more well organised in history.

The Chinese - who lived under Mongolian occupation for so long that they ended up adopting it as one of their legitimate dynasties - were nothing if not fantastic record keepers.

The History of Yuan was one of the 24 histories of China compiled during the Ming Dynasty in 1370 by the royal court, and under the direction of Song Lian.

I can't say I completely blame the source for not rooting through untranslated Chinese historical records... but suffice it to say, all the information is there, from Ghengis, to Ögedai, to Kublai, to Uskhal Khan's defeat by the Ming. If someone wants to assert the Ming Dynasty, the Islamic world, and the Vatican archives were all in cahoots since the 14th century to spin an global empire out of whole cloth... more power to 'em... but they should at least not be saying there are no records when there are voluminous records available

The History of Yuan, Full, Simplified Mandarin: http://www.guoxue.com/shibu/24shi/yuanshi/yuasml.htm

I wrote about the essential problem of the definition of "empire" in a previous post elsewhere.

Essentially, the identification of empire boils down to the acceptance of some or all of the following: socially negotiated structural definitions, self-identification, and/or recognition by others.

Which means, a case can be made by anyone whether the Mongols were an empire or not, and there is no satisfactory "universal" answer because those answers will be defined by people with a vested interest in pushing their answers a particular way.

tl;dr - The Mongol Empire exists so long as someone says it exists. Socially speaking, the broad majority of scholars and laymen say it exists. A minority, like your blog author, say it doesn't. Also, the definition of empire is not agreed upon by scholars or laymen either, so there's not really a way to compare its structural existence or not, outside of simple social acceptance.

1. ... But in Russian chronicles mongols are not mentioned.

For a midterm in a class on Russia I wrote a paper on why the Russians didn't like to talk/write about the Mongols much. According to Charles Halperin, Russian writers used ambiguous language to both protect themselves from the implications of Mongol rule and resist Mongol domination of Russian society. East Slavs had traded, intermarried, and allied with steppe nomads for centuries, and simply did not talk about these relations. They also framed militry conflicts as religious rather than political, "ignoring cooperation and idealizing conflict."

Christians elsewhere in Europe took defeat by Muslims and pagans as signs of their god's displeasure. When they failed their god, he withdrew his protection and used infidels to punish them. Either the Rus had to accept this or accept that the Christian god was not omnipotent. Russian writers "chose not to choose", and instead attempted to ignore Mongol domination. They denied they had ever truly been conquored, and continued to write as if Mongol rule never happened. When dealing with the facts of Mongol rule, they used ambiguous language to hide its causes or implications.

When recording the campaigns of Mongol conquest, Russian writers said the hordes "took" cities and principalities without elaboration. This ignored whether or not the "taken" areas were held. Similarly, they use the word "pleniti", which could be translated as "conquored" or as "plundered", which continues to leave the question of occupation unanswered. These were the same words used to describe nomad raids from the Kievan period, so based on Russian records it seems as if the same kind of raiding warfare simply continued. There are only rare mentions of princes going along with Mongol "will".

Actually I found very solid conterargument to the first statement. http://en.wikipedia.org/wiki/Imperial_Seal_of_Mongolia , it was used to send letter to Pope. So obviously Mongols had their writing system and this seal is written artefact obviously.

Side question: Google translate keeps translating something as "hamsters". Any idea what the Russian word actually means in this context?

Example:

Okay, that's understandable - hamsters protect the myth of the "" (common human or parochial) great past.