Question it doesn't look like a bad place to put a power on/off LED. It should turn off rapidly.

Are there still problems with this general purpose power supply. Give some better designs or a link to better ones.

Experimental battery chargers are good for the hobbyist, but they fall much short in technological development and will not serve for practical usefulness. Only the correct charger is designed for charging the 8.4v NiCad. But it's fine if you want to power some small circuit.

About the bandwidth of this filter, you wouldn't be able to use a resonant type. But what about a Sallen and Key filter with a gain of less than 1? A gain of 1 is okay with an input  of 20mV, maybe not for 200mV.

The resistor in this half wave rectified circuit is in series with the capacitor and battery. If you change the resistor to reduce the charging current, the battery could take a week to charge. If the resistor is in series with only the battery, it will charge in a fraction of the time.

The circuit shows a wall transformer and not a rectified AC power adaptor. So the resistor would need to be in series with the batteries. A half wave rectifier has more ripple, but the 1000uF capacitor with a 1Kohm resistor may charge the batteries slow enough. You probably have a regulated AC power adaptor. So change the resistor to a 2Kohm.

An analog motor requires an AC signal to change motor rotation direction. Their used more in appliances and their rotation direction doesn't change. Servo motors are normally digital motors. But if it's an analog motor, the frequency output of the DAC and the motors inductive reactance will require more than a power opamp.

You might try an N channel MOSFET of the same gm and Idss if you think that's the problem. Repairing electronics equipment without replacement parts isn't probably worth it.

http://www.colorado.edu/physics/phys3330/phys3330_fa05/pdfdocs/AN101FETintro.pdf

This is a good article I came across describing the developments in field effect transistors. A must see for electronics enthusiates interested in physics.

MODERATOR EDIT: Link fixed.

I have built this supply long ago, it sits in old AT case, but today I found problem, that on negative (-) supply I am having -40V DC against wall plug ground. It's not problem in general, but if you want to supply something that is for example connected to computer through USB it'll blow the fuse if not USB port of computer..
What could be causing this problem?

Serial communications between external devices often have these same sort of grounding problems which are overlooked mainly because the data is transfered reliably. Your better off using an opto-isolator.

It seems similar to the device asullivan suggested. Microswitches and motor operated cams today reminds me of a Chistmas hobby. Most devices nowdays use bytes. Programmed devices can use bits which is essential to most electrical interfaces.

You may not want to use a Sallen and Key Butterworth filter unless you want this particluar response, which can be varied. An RC filter has 20db/decade attenuation. A gain dependant on frequency opamp filter circuit can produce 40db/decade ideally. More likely it will have better filtering with added distortion.

I'll add that the inductor capacitor filter is not acting as a filter, but is dividing voltage. The frequency of the PWM is based on the frequency of the PWM.

The input voltages are very high, making the control circuit difficult to design. An ordinary opamp and PWM couldn't be used. The main pass transistor should be a low on Vds FET or bipolar transistor operated between cutoff and saturation to achieve higher efficiency. The inductor capacitor filter is absolutely necessary. Choosing the values might be done by load testing.