Is normal to have 45V at the + and - of the bridge diode?
I use 28V ac transformer.
If P2 shortcuts Pin3 of U3 with R18=U3 gets very hot;
if P2 shortcuts Pin3 of U3 with R17=U2 gets very hot, led also turns on;
I don't know too much electronics and I need someone to help me troubleshoot this power source. I've spent too many hours just to throw it away.
Your 28V AC transformer might produce 30VAC when it has low load current. 30V AC produces 42.4V peak. The rectifier bridge reduces it to 41.0VDC. Your 45V is a little too high but is OK for opamp U3 if the 10V zener diode that feeds it is not connected backwards. The cathode of the zener diode is marked with a black bar and must be connected to the positive supply. Then the positive supply pin 7 of the opamp will be +31V which is well below its maximum allowed voltage of 44V. Yours might be +35V which is also fine.
P2 is 10k, R18 is 33k and the current-calibration trimpot is about 45k. The trimpot connects to the output of U1 which is +11.2V. Then if P2 connects pin 3 input of U3 to R18 its voltage is only 11.2V x 10k/(33k + 45k)= 1.4V which is fine for U3. With a low load current on the project then the pin 2 input of U3 is 0V. Then the pin 3 input is higher than the pin 2 input so its output is high which turns off the LED and it does not limit the project's output current. U3 will be barely warm. If you are using the tiny surface-mount package then it will be obviously warm but not hot.
When P2 connects pin 3 input of U3 to R17 then its voltage is almost 0V which is fine. a load on the project that has a currernt higher than a few mA will cause the output of U3 to go low which causes 3mA in R20 which turns on the transistor driving the LED. The low at the output of U3 also causes D9 to reduce the voltage to the input pin 3 of U2 which reduces the output voltage of the project so that the output current is reduced. U3 gets a little warmer but not hot and U2 does not get hot unless the driver or output transistor is connected backwards (collector and emitter pins swapped).
You also had the output never less than 15V which might be caused by the driver or output transistor with its collector and emitter pins swapped. Then U2 and the driver transistor will get very hot.