BillO:
First, did you try using a different USB port on the PC?-Yes I tried different usb ports and also another pc.
The only things the LMV358 is connected to are the "L" led and the MOSFET. It is unlikely that the "L" LED is to blame so we are left with the LMV358 and the MOSFET. If the MOSFET is buggered, it might have taken the 3V3 regulator with it too
So, here's what I'd do. First I'd make some basic checks. I made the following measurements on a working UNO R3.
- Remove the ATmega328 to eliminate it.
-Atmega removed.
- Connect the external supply (12V) but do not plug the USB cable in. Connect digital pin 13 to ground. Pin 1 of the LMV358 should be at ~5V. Pin 3 should be at about 6V and pin 2 at ~3.3V. Pins 5,6 and 7 should be at 0V and the "L" LED should be off.
-Ground connected with 13 pin.
Results:pin 1= 4.96V
pin 2= 3.27V
pin 3= 5.46V
pin 4= 1mV
pin 5= 0V
pin 6= 95mV
pin 7= 95mV
pin 8= 4.96V
"L" Led= off
- Connect digital pin 13 to 5V. Pin 5 should be at ~5V, and pins 6 and 7 should be at ~4.5V and the yellow "L" LED should go on.
-13pin connected at 5V.
Results:pin 5= 4.96V
pin 6= 4.5V
pin 7= 4.5V
"L" led= onIf the results of steps 2 and 3 are okay and the LMV358 does not get hot, then it is at least behaving as expected and is likely okay.
-At both steps the LMV358 gets really hot as always.
Both the gate and the source of the MOSFET should be at 5V. The drain should not have any appreciable voltage on it. Mine measure ~0.5V.
-That's exactly what I measure at mine too.
The MOSFET is going to be tough to test in circuit, but with no power applied to the Uno, there should be no dead shorts between any of the pins when measured in either direction with a decent DVM.
-Mosfet seems fine with the DVM, as when you measure a good transistor or diode, so it has not dead shorts at any of the pins.
If the LMV358 measurements and MOSFET check out, the problem is likely the ATmega328.
Anyway, make these checks and let us know the results.
Dear Billo, first of all thanks for your time, that's exactly what I was mean about help.
So if these components are ok, then what cause the too much heat at lmv358?
I noticed also that the NCP1117 start gets hot too after a while, but not that much as the lmv358.
I took some measurements of this too, but it is confusing me, as I can see, when you measure it from positive or ground pin, you take different results.
Here is the results with the black connector of multimeter at ground.
pin 1= 0V
pin 2= 4.95V
pin 3= 11.33V
tab pin = 4.95V
So what's next? 
Edit: I corrected a mistype.