How damaging can a leaky microwave oven be?
More to the point, is my microwave dangerous to people in the house?


It could be, although significant power needs to be leaking for it to be dangerous: https://en.wikipedia.org/wiki/Microwave_burn


  Assuming the (900 Watt Panasonic brand) microwave is to blame for the original WiFi performance issue, could it also be possible that the leaked RF energy has physically damaged the WiFi router's transceiver circuitry?


Absolutely, although this would take significant energy. If your router has a 'site survey' feature then use that to check if the local area has suddenly sprung a number of access points on the same channels, phone apps that show the local WiFi environment are available too. 

Normally operating microwave ovens do not interfere to a noticeable extent with WiFi networks, so I would suspect that your oven is leaking to a greater-than-normal extent and that replacing it would be very wise.