There is no warm air when heater is turned on

This could be a plugged or restricted heater core and the way to find out if the overflow issue is related to the heater core is to bypass it and drive the vehicle. If it stops that issue then the heater core is plugged and needs to be replaced. If that is the case we will want to see if the heater issue is still there. If it is then most likely the blend door actuator is the issue for that. Here are some guides that will help with this:

https://youtu.be/SNRb6pSwYuU

https://youtu.be/tY6M92fA2ug

To bypass the heater core, you simply loop the inlet hose to the outlet hose from the heater core.

Let me know if you have questions. Thanks.