Undergraduate research in mathematics
To address a few of your points. 


There is no standard definition of undergraduate research. In many US universities one can graduate with a degree without doing an independent thesis, so in effect it is possible to obtain a bachelor's degree without having anything resembling "research experience". Therefore some people would be generous and count the research for bachelor's theses as "undergraduate research". For some people the phrase excludes theses but includes REU experiences, and for some only original research leading to publications count. 

It is not too unreasonable to expect that the work put into independent projects, even an undergraduate thesis that basically is just a review of existing results, can be beneficial to graduate research. 
Yes; but where did you get the idea that undergraduates only learn some linear algebra and real analysis? Combinatorics is a field in which the barrier to entry is somewhat lower, and is especially approachable to the Math Olympiad types; and discrete mathematics is certainly on the rise in the past decade or so in terms of undergraduate education. And many undergraduates do have quite a bit more under their belt than you seem to believe. 

Having worked in Europe for the past 5 years, I feel comfortable asserting that the claim 


  the level of mathematics taught in European undergrad programmes is typically higher than in the US an normally involves courses which are classed as graduate courses in the US.


is not quite true. What is true is that the mathematics in European undergraduate programs often start out at a higher level and with more rigour, but at the advanced undergraduate level the US schools usually have caught up. (Note also that the US degrees are usually 4 years compared to 3 years in Europe, so they have a bit more time to build.) Couple that with the liberal arts tradition you'll often find elective classes at the top level of undergraduate education in the US which have no analogue at all in Europe. 

While it is certainly true that good European universities offer undergraduate courses that are at the level of graduate courses in mediocre US universities, the reverse swapping European/US is also true. 
It depends on the answer to (1). The rare individual who actually did original research and obtained publishable results will likely get some bonus points when the admission committee deliberates, but it is certainly not a norm and one is not expected to have done such necessarily. 
I don't have any statistics. Hearsay suggests that original research leading to something published in a research journal is somewhat rare. (Publications resulting from REU or similar programs, or publications in undergraduate journals are less rare.)
If your master's thesis contains original results, then you are likely more than on par with the typical applicant to US PhD programs, in terms of research experience.