A poor seal or a split.
A damaged heater core will cause coolant or water to leak onto the passenger floorboard.
water pump, oil, water, engine coolant....main cause would be engine coolant, just check to see if you have a leak.
Leak in heater core
No, low coolant would cause the opposite.
If it is water it is probably the condensation drain plugged up with crud. If it is engine coolant it is probably the heater core leaking.
Many things could cause an engine to shake. One, for example, would be low coolant or oil levels.
Coolant Temperature Sensor
No, if the coolant circulates too quickly it will not have time to cool down and re-cool the engine. The coolant would become extremely hot, and cause your engine to overheat and thus causing possible engine damage.
I believe that indicates your engine coolant level is low or you have lost your coolant
I would say yes, if oil has mixed with the coolant, that's the most probable cause.
If we are talking rain water not engine coolant I would suspect the drains under the cowl are plugged up with debris (leaves and such). If it is engine coolant then I would suspect the heater core is leaking.
engine coolant getting into an engine cylinder from a bad head gasket , cracked or warped head