Calculating nominal GDP I would like to understand one detail of how nominal GDP of a country in a given year is calculated. It seems to me that the GDP in each country is first calculated in the national currency and then is converted to dollars. Is this correct?

If yes, how can one make this conversion if a currency can weaken of strengthen itself significantly with respect to dollar within a year?

Here is the list of such GDP's per country per year in US dollars:
http://en.wikipedia.org/wiki/List_of_countries_by_GDP_(nominal)

For example take Russia, GDP in 2013 was 2,096,774 million dollars and in 2014 it is 2,057,000 million dollars.

Now, we know that the rouble weakened significantly in 2014, (almost twice). So it looks like Russian GDP in dollars should have dropped significantly as well. But it had not done so. Why?
There is no need to complicate things here - just try to find annual exchange rate used by IMF. Worldbank's data are here. LCU stands for Local Currency UnitsYou may see a drop of Rub/$ rate from 31.84 to 38.38. (Drop of Ruble, of course.)
National statistics provides input figures for international accounts. According to Rosstat, Russian nominal GDP grew in the Y2014: 


Y2013 - 66,190 bil. Rub, 
Y2014 - 71,406 bil. Rub. 



Using  NGDP dollar evaluation from your post, annual exchange rate for Y2013 is 31.5 Rub/$; in Y2014 rate is 34.7 Rub/$, which makes some sense. But indeed Rub/$ rate was somewhat closer to 38.5 (see here). So, IMF figure for the Y2014 from your wiki reference seems to be more correct: bil. $1,857,461 for Russian NGDP. (do not forget about Crimea correction to appreciate the difference).
So, in fact Russian nominal GDP did drop significantly in current dollar terms in the Y2014. 
Imho, PPP is not applicable for your question as Russian GDP by PPP is of different magnitude.