The one thing you're right about is that Slovakia didn't play well. As for the rest:
All teams may have faced equal difficulty in getting their NHL players in the qualification round, but because Slovakia has so many more NHL players than, say, Belarus, it faced a greater handicap in reaching its potential as a national side. So, no, the playing field was not even.
Nor is it absurd to suggest a connection between socio-economic conditions and the performance of national sports teams. Your example of NHL millionaires and unemployment is specious: a young nation which is struggling with issues like national identity, and whose citizens are still toiling through transformation, is bound to produce sports teams that are tentative and lack confidence. At least, this is the opinion of psychiatrists and sociologists who live here. If you have greater insight into the national psyche you should tell us what it is.
Finally, I don't think Slovaks play the victim (unless you aimed the comment at me, in which case I guess I'll have to check that mirror you mentioned). On the other hand I think they face some handicaps compared to western societies, and stand more in need of a boost, whether from sports or whatever, than citizens of more developed countries.