Sub-zero temps. last 2 nites. vehicle starts/runs fine except for radiator fan that is running at high speed and not allowing engine to warm up.

My information shows that the 2001 tahoe uses a fan clutch and does not any electric fan. There could be exceptions, but if this is an add on aftermarket fan, there would be some type switch with a relay or fuse wired in the system. First make sure that the radiator is not partly restricted. Using a inferred thermometer check to see is all the radiator tubes are the same temp. Most of the time, the lower tubes are cooler because they are restricted. Also make sure the fan clutch is good. After driving it and the engine is at running, shut the engine off and watch how many times the fan rotated. It should stop after several revolutions.

Take it to a qualified repair shop that has diagnostic information (shop manual ) An has a professional scan tool . Changed what sensor ? Coolant temp ? You don't guess an just replace parts . Doing a full vehicle scan ,checking for DTC'S in all the modules on the vehicle.

ENGINE HOT A/C OFF
Refer to Cooling System Description and Operation
Do the radiator cooling fans work ?
The engine cooling fan system consists of 2 electrical cooling fans and 3 fan relays. The relays are arranged in a Series/Parallel configuration that allows the engine control module (ECM) to operate both fans together at low or high speeds. The cooling fans receive positive voltage from the cooling fan relays which receive battery positive voltage from the underhood fuse block.
During low speed operation, the ECM supplies the ground path for the low speed fan relay, which is Fan 1 Relay, through the low speed cooling fan relay control circuit. This energizes the low speed fan relay coil, closes the relay contacts, and supplies battery positive voltage from the Fan 1 Fuse through the cooling fan motor supply voltage circuit to the left cooling fan. The ground path for the left cooling fan is through the Series/Parallel relay, which is Fan 2 Relay, and the right cooling fan. The result is a series circuit with both fans running at low speed.
During high speed operation, the ECM supplies the ground path for the low speed fan relay through the low speed cooling fan relay control circuit. The ECM grounds the high speed fan relay, which is Fan 3 Relay and the Series/Parallel relay through the high speed cooling fan relay control circuit. This energizes the Series/Parallel relay coil, closes the relay contacts, and provides a ground path for the left cooling fan. At the same time the high speed fan relay coil is energized, the relay contacts close to provide battery positive voltage from the Fan 2 Fuse on the cooling fan motor supply voltage circuit to the right cooling fan. During high speed fan operation, both engine cooling fans have their own ground path. The result is a parallel circuit with both fans running at high speed.
When the requests for fan activation is withdrawn, the fan may not turn OFF until the ignition switch is moved to the OFF position or the vehicle speed exceeds approximately 10 mph. This is to prevent a fan from cycling ON and OFF excessively at idle

DTC P1258: Engine Coolant Overtemperature - Protection Mode Active

No the cooling fan should run continuously.

If there is bubbles occurring even inside the top part of the radiator (where you have the coolant put in), you may have a leak on your head gasket. You may be loosing coolant through the engine.

First replace your Thermostat, and then have your Electric cooling fan checked.

Try getting a scan at Autozone, Advance, or Oreillys autoparts. They will do it for free. Or if you just want to buy a few parts, observe if the cooling fan even comes on. There are 2 temperature sensors on most Ford's and one makes the dash gauge work and the other makes the Electric cooling fan work. Just have the car a little warm and when you get the dash gauge to move, disconnect the temp sensors one at a time until you find the one that controls the gauge. The remaining sensor controls the cooling fan. Remember, the gauges only work when the engine is running.

If your cooling fan still does not work, ask the parts guys how many speeds your vehicle is suppose to have on its fan. Usually, when the A/C is turned on, high speed kicks in; for sure when you have overheating, it should be on high speed. The Electric cooling fan has a Relay that can be bad. The fan motor can also be bad.

The loss of Antifreeze in the system can be from boil over. As long as the car runs decent, there is no reason to expect a blown head gasket. With the Radiator cap off, you may be able to see bubbles on the surface of the radiator if a head gasket is allowing compression to get into the cooling system.

Just start with the easiest things like a Thermostat, then go to the next solution and work your way through the possibilities. I hope my comments are useful to you.

It is located under the hood between the radiator and the front grill. You will have to remove the platic cover between the radiator and grill to see it.

Hello,
Most often that is caused by a bad thermostat.

How to test the thermostat in most vehicles;

Make sure the heater is off, start the engine and let it run for 6 to 8 minutes, ( this will allow the
engine to reach, " normal operating temperature "), from the inside of the vehicle turn the heater
temperature on high and turn the fan speed on high.
1. Listen very closely for a click sound from the engine, ( most thermostats make a click when
they open )
2. Watch your temperature guage, it should begin to lower as the coolant starts to circulate.
3. Usually within a few minutes the radiator fan will kick.

What the thermostat does;

It seperates your coolant into two parts. The first part is in your engine, the second part is in your
radiator and the coolant reservior. When the coolant in the engine reaches the temperature rating
on the thermostat the thermostat opens to allow the two parts of coolant to change places. Now
the cooled coolant in the radiator and reservior flows into the engine while the heated coolant flows
from the engine into the radiator and reservior to be air cooled by the radiator fan.

I hope this helps you.

Take a look through your front grill and see if there is alot of bugs,dirt,etc. blocking the cooling fins of the condensor and radiator. If so try rinsing them with a water hose, or air pressure. Some cleaners work well also,like " Simple Green " or gunk degreaser. Catch the fluid in a container and dispose of properly.

Have your radiator flow checked you may have a clogged radiator.

If the engine is not over heating then it is doing its job. The constant cycling is based on the temp of the air traveling through the radiator and the a/c condenser not the outside temp so much. The start up high rev is just how these clutches work in ford fans. It sounds like it is operating correctly.