Can an Arduino Pro Mini interface with a full-sized WiFi shield?
I would like to know if they can be connected and, if so, how.


Yes.


  but can we place the shield on the breadboard


No, the pins are misaligned and won't plug in.


  and use jumper wire to connect the necessary pins so the Mini can use the WiFi


Yes - connect the jumper wires directly to the WiFi shield pins.


  If so, how would we know which pins are used, 


Read the manual or schematic for your WiFi shield. It probably uses the SPI pins plus a few more.


  and would we need to provide additional power?


It will need a power connection, yes - though "extra" power? Not sure what you mean.