"I want to eat very much" would normally be interpreted as referring to the quantity I want to eat, and not the level of my desire to eat. In that sense it's different from "I really want to eat" and from my interpretation of the Ukrainian sentence. "I very much want to eat", which is not accepted, would be a somewhat unfashionable, but still correct, way to express "I really want to eat".
Maybe I should have mentioned this in my original report, but I am a native speaker. Also, you might be interested in Google's book search results for the phrase, which confirm that the phrase is generally used in the way I described.
EDIT: Removed garbled link to avoid confusion.
In the Google Books results, it seems "very much" is only used as a quantity in the negative or comparative sense, which I will admit is a case where "very much" can be used as a quantity. However, I still believe that "I want to eat very much" and "I really want to eat" are synonymous.
There's something a bit bizarre going on with the link. When I click now I don't get the book results, even though I copied the URL from the address bar after such a search. You can still get the results of that search by going to the ngram viewer, entering "eat very much" into the text field, searching, and then viewing the sources. I looked at the context for the first 20 results. It's about 75% negative, 25% positive. The positive uses all seem to refer to quantity.
Duo doesn't like nested comments beyond a certain level, so I'll have to reply to my own post rather than yours. I never denied that "very much" can mean "really". In fact, if you reread my posts you'll see that I suggested that "I very much want to eat" should be accepted as a translation, which uses "very much" in precisely this way. My problem with "I want to eat very much" is the word order. Because "eat" separates "very much" from "like" it is more natural to interpret "very much" as modifying "eat" rather than "like". "Very much" can indicate either quantity or intensity generally, but in the case of "eat" only the quantity interpretation makes sense, while in the case of "like" only the intensity interpretation works.
jgstcd, I understand your syntactic argument for wanting to interpret "very much" as referring to quantity, but I would still interpret "I want to eat very much" as synonymous with "I really want to eat", simply because "very much" never occurs as a positive quantity under any circumstances in my dialect of English (western US). I'm willing to accept that this may not be the case for other dialects or registers, however.
It could be said, but it would have a negative implication, i.e. the questioner expects that this person has insomnia or isn't able to sleep much. The more common way in my speech would be "Does he sleep a lot?", although this has more of a positive implication - the questioner expects that the answer is yes.