Did anyone do a research on corelation between honesty and success, establishing individual's honesty through a provocative experiment and comparing it with his/her salary amount, satisfaction of achievement etc? Or anything close to that?
No provocative experiment needed. Some of the major taxonomies within personality psychology have well-validated measures for honesty, and have been used in massive samples that could already address this question. I would start by doing a search on Google Scholar for those measures and the types of success you mention. E.g.,
Notice that the trait most often associated with success is Conscientiousness, which may include an aspect of honesty:
To add to the existing answer, honesty-humility is a major factor in the HEXACO framework of personality.
There is a comprehensive list of papers using the HEXACO framework here organised under a wide range of headings: http://hexaco.org/references
My sense of the literature is that honesty-humility in the HEXACO framework is related more to ethical behaviour and lower levels of deviant behaviour (e.g., counterproductive work behaviours, criminality, deviance, etc.), as well as lower levels of self-interest and egotism.
I don't think it has a major direct relationship with success and achievement, at least empirically. But I haven't reviewed the literature.
Theoretically, deviant behaviour would often reduce life success. Getting fired, damaging your reputation, going to prison, getting expelled from school, and so on. However, also, some degree of self-interest may also assist in personal achievement. Furthermore, it may just be that it's not the most relevant dimension of personality: i.e., conscientiousness and abilities are probably more related to achievement.
(This answer your question only partially).
Honesty and success are made of many things. Quite a bunch of studies found interesting correlations between honesty and success at the collective/cultural level. (They measure the success in term of wealth, trust, quality of the institution, rights, etc.).
Just a sample to give an idea:
● Edward C. Banfield (1958) shows some correlation between many aspects of honesty and success (he wrote a book about it The Moral Basis of a Backward Society).
● Arrow (1972) & Fukuyama (1995) found that the level of trust in a society strongly predicts its economic success.
● Guido Enrico Tabellini (an Italian economist) find that "there is not just one institutional failure. Typically, the countries or regions that fail in one dimension also fail in many other aspects of collective behavior."
● Inglehart (1990, 1997) analyses of a broader range of societies, and argue that cultural values fundamentally drive economic performance and democratic stability.
● Landes, D. S. (1998) & Rogers D. found that people's value shape their wealth (they wrote a book: The wealth and poverty of nations: why some are so rich and some so poor)
References: (you'll find many others in the references of their studies).
● Banfield, E. C. (1958). with the assistance of Laura Fasano Banfield. The moral basis of a backward society.
● Arrow, K. J. (1972). Economic welfare and the allocation of resources for invention. In Readings in industrial economics (pp. 219-236). Palgrave, London.
● Porta, R. L., Lopez-De-Silane, F., Shleifer, A., & Vishny, R. W. (1996). Trust in large organizations (No. w5864). National Bureau of Economic Research.
● Tabellini, G. (2008). Presidential Address: Institutions and Culture. Journal of the European Economic Association, 6(2/3), 255-294. Retrieved from http://www.jstor.org/stable/40282643
● Inglehart, R. (1997). Modernization and postmodernization: Cultural, economic, and political change in 43 societies. Princeton university press.