Is Salmonella from eggs a US-only problem?
The short answer is "No," it's definitely not a US-only problem.  Since the question makes comparisons between the US and Europe, I will focus mainly on those two areas.



Unfortunately, it's hard to estimate illnesses from eggs only, because frequently Salmonella outbreaks from eggs are due to how they are handled in the kitchen and what other foods they may contaminate.

If we just look at Salmonella cases overall, the most recent figures I could find from the EU (2014) list:


88,715 reported (confirmed) cases, 23.4 cases per 100,000 population
9,830 hospitalizations
65 fatalities


The most recent figures I could find for the US (2013) list:


7,307 reported (confirmed) cases, 15.2 per 100,000 population
2,029 hospitalizations
30 fatalities


(Note: these figures appear only based on a sample of the US population, so I'd pay attention to the incidence rate per 100,000 when comparing with the EU numbers.)

The problem, of course, is that in both the EU and the US the number of reported cases is only a small fraction of the total.  A major study from the EU estimated that only 1 in 58 cases of salmonellosis is reported, which would suggest an annual incidence of perhaps 5 million cases.  The CDC report from the US linked above estimates approximately 1.2 million cases every year.

Given that the population of the EU is around 500 million and the US is about 325 million, the incidence of Salmonella poisoning appears to be roughly double in the EU compared to the US, but again these are only estimates.  (Also, it's notable that that last study says the incidence rate in various EU member states likely varies between about 16 per 100,000 and 11,800 per 100,000, which is obviously a huge range.  Some countries are much more safe than others.)

Now, the question is: how many of these cases can be linked to eggs?  Most studies suggest that eggs are the most common carrier of Salmonella leading to illness.  From the above EU report: 


  Salmonella was rarely found in table eggs, at levels of 0.3% (single samples) or 1.0% (batch samples).  The most important source of
  food-bourne Salmonella outbreaks was, however, still eggs and egg
  products.


In the US, one study from 2000 estimated somewhere around 180,000 illnesses annually from eggs, but another report estimated that the proportion of foodborne illnesses associated with eggs decreased from 6% in the late 1990s to 2% in 2006-2008.  This was all before the new egg safety rule passed in 2010.  (That link estimates 79,000 Salmonella cases annually associated with eggs before the rule.)

Compare that to the EU, which estimates 5.4 million cases of Salmonella in 2010, about 17% of which were likely attributable to eggs, or 928,000 cases in 2010.  Given the differences in population, that suggests that Salmonella cases caused by eggs were about 7 times more prevalent in the EU compared to the US in 2010.  That's the most recent year I could find decent figures for eggs alone to make a comparison.  (Note that incidence various significantly across the EU member states. Those countries that have stricter policies, like the UK's vaccination policies, have much lower numbers.)

Estimates of the fraction of eggs in the US contaminated with Salmonella have often suggested a number around 1 in 20,000 (or 0.005%), based on a study from 2000.  But that was before the new egg safety rule was introduced, and the numbers are likely somewhat lower now (given more frequent monitoring of infected flocks and their removal from production).



With all of that said, I sense in the question and the links provided a skepticism about US egg policy in general.  Just to clarify a few points related to those sources:


Chickens are much more likely to be vaccinated in the EU compared to the US, but the percentage of infected chickens still varies significantly from country to country in the EU.
Improper egg washing is a problem, but guidelines for appropriate temperatures, etc. have become much more strict in the US over the years.
Egg washing is most definitely NOT just done for appearance.  The US successfully lowered its Salmonella rates significantly earlier than many European countries through egg washing and refrigeration.  Similar gains in safety have only recently been attained in Europe mostly through vaccination instead.  The vast majority of Salmonella cases from unwashed eggs come from shell contamination.  Without washing eggs, the only way to ensure safety is to make sure the exterior isn't contaminated in the first place (which means ensuring chickens aren't infected in the first place, e.g., through vaccination).
The last report I linked from the European Food Safety Authority states in its abstract: "An effective way to minimize any increase in risk during extended storage is to keep the eggs refrigerated both at retail and the household."  The practice of not refrigerating eggs in the EU is likely a major contributor to the higher incidence of Salmonella cases mentioned above.  In the US, refrigeration is required partly because of the washing process as well, which does increase shell porosity somewhat.