Bad water pump, the seal is gone. Get a new water pump.
water pump, oil, water, engine coolant....main cause would be engine coolant, just check to see if you have a leak.
No, low coolant would cause the opposite.
Many things could cause an engine to shake. One, for example, would be low coolant or oil levels.
Unless it's something extraordinary, a leak in the heater core is about all that could cause it. It may be that condensate from the AC core is draining inside the cab, but that is unlikely. If it's a heater core leak, the fluid on the floor will be engine coolant, and if you have anti-freeze in it, you'll have that on the floor. And your engine coolant level will be dropping. It is possible that the engine may be overheating for lack of coolant.
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.
A poor seal or a split.
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
No. It would cause the coolant and oil to mix and it would look like a Milkshake drain oild and check it.