I booked a flight on the 19th of November, but my Russian visa expires on the 22th of November.
Would there be any problems to board my flight and enter the country?
No, if you have a valid visa and enter the country in time, and you didn't exceed the number of entries and number of days available in Russia, there wouldn't be any problems. However, you still need to leave the country in time, without staying in for too long before your visa expires.
As per rules, there should not be any issues entering the country with valid visa (if you are in good terms with previous travel and stay)
I have entered the US with only 3 days available on my visa.
Border security will definitely ask questions about return flight or extension of visa, that you should have a SOLID answer (may be a return flight ticket, visa extension application, or similar stuff).