It is currently 16 degrees with freezing precipitation. This expected to continue for the next 2 days. Should I switch my thermostat to emergency heat? And if so, is it okay to run it on that setting for the next 48 or so hours?

Heat pumps are great in certain environments, 16 degrees is not one. Is there heat in 16 degree air? Yes but outflow will overcome inflow. Sure you can use emergency heat but you will pay the cost in $$$. Better to let heatpump run and add an extra sweater. You are finding out the shortcomings of a heat pump at 16 degrees.