How can I create a map of all paths of flights incoming to an airport in a given day?
Well you didn't state which airport you are interested in but obviously you only want data for a single airport.

You might want to look into a noise abatement reporting system. For instance, London Heathrow maintains the great website Webtrak (https://webtrak52.bksv.com/lhr4) to communicate with the pubic about changing SIDs, STARs and noise levels. (They already have much of the TMA over the heavily populated area, and they want to add another runway)