I've been doing an experiment using liquid nitrogen, and am aware that pouring it into different containers gives a chance for other gases to liquify and contaminate it; what's a good estimate for the amount of other gases in the flask in such a situation?
That depends mostly on the time the liquid is exposed to the other gases and the container you are keeping it in. A quantitative answer is hard to give though.
In a clean glass dewar a little bit of evaporated nitrogen will hover over the liquid phase and will slow down the condensation of other gases into the liquid phase. In a open container with more airflow you will get contaminations with oxygen, water and CO2. I have observed that small ice crystals and dry ice float on top of liquid nitrogen, the exact amount depends again on your conditions: Just breath out just above the liquid and you can see it condensing/freezing.
Other than that you might have dirt that comes from your dewar as larger ones are hard to clean thoroughly but this can be filtered out easily with a fine paper filter.
One way to keep the liquid without contaminations is to first clean your transport vessel, evacuate it, fill it just with liquid/gaseous nitrogen and then keep it sealed with a pressure release valve under a bit of overpressure (actually two or three valves as required by law). The release valve prevents other gases from coming in and also reduces your rate of evaporation, as the nitrogen can be kept a slightly higher temperatures.