Can RIVER/DELTA run a heater/pump/fridge? And for how long?

This depends on the specifications(watts) of the device. You can contact support@ecoflow.com with the specifications(watts) of the devices you want to run and the CS team will help you suggest the best power station to run it.

Extra Battery work logic

To charge or discharge the EB, it has to be connected to the power station.
The charging logic for the power station+ EB is that the one with the lowest voltage level will charge first then when both the power station+ EB are at the same voltage level they will continue to charge together.

As for discharging the one with the highest voltage level will discharge first and till both the power station+ EB are at the same voltage level they will continue to discharge together.

https://ecoflowtechhelp.zendesk.com/