Check engine light is on. When at a stop light, the truck will shake and eventually smoke some. We are currently replacing the battery and a tire, and plan on taking it in but are curious what the potential problems may be with the shaking and smoking? (It does not say that it is over heating though.)

Most likely a tire pressure monitor light. Check pressure in your tires. If you fill them up to the correct pressure and the light goes out, probably after driving a bit, the mystery is solved. By the way, the sensor is not part of the tire. If the sensor is bad, the only way to stop the blinking is to replace the sensor that is bad. A bad sensor won't hurt anything. You just won't know if one of the tires is low and you will have to check such yourself.

If the SH-AWD indicator on the instrument panel stays on, and the multi-information display shows an SH-AWD message, there is a problem with the system. Your vehicle still has normal front-wheel drive with vehicle stability assist (VSA), but does not have the advantages of SH-AWD. No check engine light because this has nothing to do with the engine.

Most reputable tire shops have the tool to access the tire monitor system. The tire pressure sensors in each tire have a battery that goes dead eventually.

The SH AWD uses electromagnetic clutch in the back drive axle to change the power distribution to the rear wheels. It would have nothing to do with the P0171 error. What TPMS valve stem was bad? Because this could be a clue to what the problem is. If a rear tire is having issues it could cause the sensor to make adjustments to the system to compensate for the tire. The person that changed the TPMS might have also caused the issue when they worked on the tires.

I would be checking for the main lead to the battery /starter to ensure that it has not burnt though on the exhaust or has not worn through and been shorting out. The clicking noise would ne the solenoid moving but the current is not getting to the starter. Check the condition of the battery. Shorting of the cable will take the current away from the electrics so the engine would stop. The battery will have to be load tested to ensure that it has not been damaged.

Well the smoking fuse would mean the circuit between the alternator and battery is overloaded just like in a house.
Either a short to ground or a component drawing too much current.

Is the Alternator charging. Could have a short drainning battery also. Check fuses.

Blue smoke is engine oil burning in the combustion chamber. White smoke would be coolant and black smoke is too much fuel. Try replacing the spark plugs to get better fuel mileage, they are most likely fouled due to oil burning.

+++ been told to fix blue smoke...piston rings gone .

Piston rings, valve seals are a good guess for cause of blue smoke. Engin oil additives for high-mileage engines can help the blue smoke problem... Pep Boys and the like -- many different products -- just make sure it's for stopping smoke, not cleaning engines...

+++ squealin....fan belt gone.

Has the alternator belt been tightened? If not, this could be causing both symptoms: dead battery & squealing

+++ battery light on...alternator gone.

Again, this could just be a loose belt, and most likely is the problem...

Must have codes for check engine light cleared with scan tool. Check all tire pressures and road test up to 35 mph. Tire message should clear if low tire was problem. If light for tires remains then possible radio frequency sensor problem. Sensor replacement may be necessary. Sensors have alkaline non-serviceable batteries. If battery is dead sensor will have to be replaced.