My car stays in first

Squeaking noise could be from the drive belts. Have them checked.

White smoke and necessity to replenish the coolant indicates a possible bad head gasket. Is the recovery tank depleting as well or only the radiator is depleting?

Transmission not upshifting could be a failing transmission, out of adjustment throttle pressure cable or insufficient transmisison fluid.

When belts ages, they tend to harden and produce squeaky noise. If there is any slack in the tension, squealing noise would occur ocasionnaly, especially during startup or sudden accelerations.

If only the radiator is depleting whereas the recovery tank is not and might be overflowing instead, it could be a bad head gasket, leaks between the recovery tank and radiator or a bad radiator cap. If both are depleting, you have a leak in the system and a pressure/leak down test should be able to detect the leak. A visual inspection might reveal something, look out for signs of staining due to presence of coolant, especially around hose joints.

If there is any problem with the valve body, flushing would not help.
Does accelerating and releasing the accelerator pressure allows the gears to up shift? If not, you have a fault with the valve body which requires the valve body to be serviced. Does manually shifting of the gears result in upshifting?
Does the D4 indicator light blinks while driving?